Edit: Also just to preface this by saying: I am not a lawyer, and in a contemporary setting this is a legally complex topic.

While I'm not familiar with the legal authorities under which the Klan was "designated" as a terrorist organization in 1870, or what the implication for such a designation for law enforcement purposes would have been at that time, I can answer the question for today, or at least within the bounds of what the 20 year rule allows.

The contemporary power to designate a group as a terrorist organization was given to the executive branch (specifically the Secretary of State) by the Antiterrorism and Effective Death Penalty Act of 1996.

That act:

Amends the Immigration and Nationality Act (INA) to authorize the Secretary of State, in consultation with the Secretary of the Treasury (Secretary) and the Attorney General, to designate an organization as a terrorist organization upon finding that the organization is a foreign organization that engages in terrorist activity and such activity threatens the security of U.S. nationals or U.S. national security.

Sets forth provisions regarding: (1) procedures for such designation, including notification to specified congressional leaders, and the freezing of assets; (2) creation of an administrative record and the handling of classified information; (3) the period of designation; (4) revocation by Act of Congress, revocation based on a change in circumstances, and the effect of revocation; (5) use of the designation in a trial or hearing; (6) judicial review of such designation.

(Sec. 303) Sets penalties for knowingly providing, or attempting or conspiring to provide, material support or resources to a foreign terrorist organization. Requires any financial institution that becomes aware that it has possession of, or control over, any funds in which a foreign terrorist organization or its agent has an interest, to retain possession of or maintain control over such funds and report to the Secretary the existence of such funds, with exceptions. Establishes civil penalties for knowingly failing to comply with such provision.

While I have to confess to being much more familiar with these sections of the US code as they have since been amended by the Patriot Act of 2001 and subsequent legislation, this provides the basic answer:

The authority to designate a terrorist organization as such is limited strictly to foreign terrorist organizations.

While the US Code does have statutes outlining crimes that apply to domestic forms of terrorism, they don't involve or require "designation."

Designation comes into play, for example, when the Treasury Department announces that it is sanctioning individuals who are affiliated with a designated terrorist organization, or when an individual is charged with providing "material support" to such groups.

Why do these laws apply to Foreign Terrorist Organizations (FTOs) with no equivalent domestic authority? The fact is that these sweeping powers, if applied in a domestic setting, would raise serious civil liberties concerns, particularly around free speech and free association. Pledging allegiance to an FTO is often sufficient to charge someone with material support to that group (the material support in this case being your own allegiance.) (Edit: As the Lawfare piece mentions, there are some material support charges that apply in a purely domestic setting, but my understanding is that they are substantially narrower than what applies to an FTO.)

This piece in Lawfare is not "neutral" (the author has an opinion on the question) but it does go into great detail about the various foreign and domestic terrorism codes are in US law and why it might be a bad idea to apply them to domestic groups:

https://www.lawfareblog.com/should-we-create-federal-crime-domestic-terrorism

Edit 2 One other point I'd like to note is this particular provision, which is echoed in other statutes relevant to defining an FTO:

the organization is a foreign organization that engages in terrorist activity and such activity threatens the security of U.S. nationals or U.S. national security.

Next to the question of why the US does not designate domestic groups, it is frequently asked, often for political purposes to criticize the United States, why the US doesn't designate all terrorist groups or why such-and-such particular group that threatens the stability of that country has not been designated (i.e. implying that the US is sympathetic to terrorists by failing to designate.) The answer is that the Secretary of State has to find, specifically, that the group threatens the security of U.S. nationals or U.S. national security. Many terrorist groups, having a more limited set of objectives, fail to meet that threshold and so cannot be designated. One recent but longstanding example of this is the debate over whether or not to designate the Muslim Brotherhood as an FTO. While this example is within the 20 year rule, it's a good example of the statutes in play. Lawfare, once again, with a good analysis of that question.

So a small correction here. The Klan was not designated a terrorist organisation in 1870 (at least not in the way we would consider it) and it was not outlawed. However, a number of measures were brought into place to try and combat it by the federal government and the group eventually faded away or its members took up with other so-called white line organizations (i.e. white supremacist groups).

So the Klan was formed in Pulaski, TN in late 1865 or 1866 and we are slightly unsure of why it was initially formed. For many years, scholars assumed it was formed by six former Confederate officers (the Jolly Six) who were bored with rural life during Reconstruction. I say assume because a lot of this is drawn from the only account of this period, which was written by John Lester and D.L. Wilson in 1884. Lester was one of the Jolly Six so recently historian Elaine Frantz Parson, in her recent and excellent discussion of the Klan, Ku-Klux: The Birth of the Klan during Reconstruction (UNC Press, 2015) has questioned whether we should trust this source of information, which was published in 1884 and which would have tried to deflect and say the group only became violent when they were forced to by radical Republicans and freedmen attacking whites. Parsons has re-assessed this origin story and piecing together what little information we have has demonstrated that violence and terrorism was their object from the start. Its not entirely certain since there is not much information, but this violent group eventually begins to spread beyond Pulaski and into surrounding states.

The popularity of the group in other areas eventually leads to the formalization of the group and to attempts to create a hierarchy and a military structure to lead the group. There’s information online, but in essence in April 1867, supposedly at the Maxwell Hotel in Nashville Tennessee, the Klan writes its constitution and designates its officers. We know very little about what happened here since again, the only information we have comes from Klansmen, but its supposedly where Nathan Bedford Forrest became Grand Wizard. Moving on, we believe the leadership begins trying to spread the Klan more formally, but by this point, Klan chapters are cropping up across the South as news spread of their weird rituals and costumes and of how effective their terror tactics are.

For years, we assumed that there was little coordination between Klan groups in these early years after 1867, and that this is what led to such brutal violence. This is somewhat supported by the fact that each Klan only sort of follows general guidelines and targets specific people. You can see this reflected in their costumes, which are not at all how they’re represented in films. They vary immensely from just a bag over the face, to elaborate and multi-coloured outfits. Again, Elaine Frantz Parsons in her excellent 2005 article “Midnight Rangers: Costume and Performance in the Reconstruction-Era Ku Klux Klan,” has shown that many Klansmen even dressed up as women, as Native Americans, as beasts (real and imaginary) or even in blackface to look like freedmen, in an effort to disassociate themselves from their identity as white men. Basically, they dress up so they can commit brutal acts of violence unbecoming of white Southern gentlemen. This variation in costumes, theatricality and violence suggests there is little communication or coordination between Klan chapters. In fact, the records of the Joint Select Committee (more on this later) have many interesting accounts of separate Klan groups encountering each other while on a raid and standing off before realizing who each other was. Okay, so scholars always thought the Klan was very loosely associated during this early period, and really its more appropriate to term it the Ku Klux Klans because of how many groups there are. Recently though, Bradley Proctor managed to decipher a bizarre document in South Carolina that demonstrates some Klan chapters were in fact communicating with each other and central leadership – See “"The K. K. Alphabet" Secret Communication and Coordination of the Reconstruction-Era Ku Klux Klan in the Carolinas”

However, this is not that important, as the majority seem to be acting independently, and some in the mountains of North Carolina where there were very few freedmen even use the Klan to protect their moonshining operations from federal revenue collectors. The brutal violence occurring across the South and the unstoppable spread of the movement forces the formal leadership of the Klan to issue General Order No. 1 in 1869, the only official order of the group, which calls for Klans to disband. But by this point, they can no longer control anyone, and the order is widely ignored. So many Southerners have taken up the mantle of the Klan, and just as importantly, people were referring to any group of armed white supremacist terrorists as Klan, that the order keeps growing. A number of white-line groups such as the Pale Faces in Tennessee get referred to as Klan simply because of what they are doing.

This is getting a bit long now, but just to answer the specific question: what Congress does is outlaw the Klan’s crimes, namely the congregation of people to deprive others of their civil rights (particularly the right to vote). The three Ku Klux Klan Acts are quite vague precisely so that they can encompass a broad range of white supremacist groups. They are able to do this after having collected hundreds of pages of testimony of Klan victims by the Joint Select Committee to Inquire into the Condition of Affairs in the Late Insurrectionary States, which brought to light and brought official evidence of how widespread and brutal Klan violence was (full documents at http://onlinebooks.library.upenn.edu/webbin/metabook?id=insurrection1872)

These Ku Klux Klan Acts allow Congress to do such things as suspend habeas corpus, and they do so in one of the worst Klan-stricken areas in the country, the South Carolina upcountry. Here, Major Lewis Merrill and a detachment of U.S. Cavalry are sent to investigate and put down the violence. They eventually round up Klansmen and begin prosecuting them. This spooks a lot of Klansmen across the U.S. who realize they have been drawing too much attention to themselves, and many just stop their activities until stuff dies down. Richard Zuczek has argued that the prosecution of these South Carolina Klansmen did little to stop the Klan, namely because many of those who were rounded up were not prosecuted properly (it was very difficult to prove they were involved) and the government did not have the resources or the political will to carry out these prosecutions. After all, suspending habeas corpus was big deal. So it was really the threat of federal intervention that dissolves the Klan, rather than the actual prosecution. Klan activity begins to die down after 1871, and practically fades by 1872

But the situation doesn’t end there. The specific markers of the Klan, such as the midnight raids, the elaborate costumes and theatrical titles stop being used, as they draw too much attention. But the men who participated in these groups simply join other groups and continue carrying out acts of violence. If anything, these new groups are more threatening, since they are more regularly organized, have better military equipment, leadership and coordination, and basically act more openly. A good example of this would be the White League of Louisiana, that cartoonist Thomas Nast basically said was the ghost of the Klan in new form in this well-known 1874 cartoon https://www.loc.gov/resource/cph.3c28619/

So the key point and answer to your question is that the Klan was never outlawed or designated a terrorist organization because it never really was an organization in the formal sense of the term. What they do outlaw are the specific activities of the Klan, namely the act of conspiring to deprive someone of their civil rights.

Also, happy to answer any follow-up questions. Further reading: Elaine Frantz Parsons, Ku-Klux: The Birth of the Klan during Reconstruction (University of North Carolina Press, 2015)

Elaine Frantz Parsons, “Klan Skepticism and Denial in Reconstruction-Era Public Discourse,” The Journal of Southern History Vol. 77, No. 1 (February 2011), pp. 53-90 -

Elaine Frantz Parsons, “Midnight Rangers: Costume and Performance in the Reconstruction-Era Ku Klux Klan,” The Journal of American History Vol. 92, No. 3 (Dec., 2005), pp. 811-836 -

Michael J. Martinez, Carpetbaggers, Cavalry, and the Ku Klux Klan: Exposing the Invisible Empire During Reconstruction (Rowman & Littlefield Publishers, 2007)

Richard Zuczek, “The Federal Government's Attack on the Ku Klux Klan: A Reassessment,” The South Carolina Historical Magazine, Vol. 97, No. 1 (Jan., 1996), pp. 47-64