According to Wikipedia, Congress did not count the electoral votes of Tennessee or Louisiana in 1864 even though they had cast votes. I have a few questions:
(1) Well, "part of the Confederacy" is begging the question. There were several legal theories pertaining to the states that had seceded. Charles Sumner held the "state suicide" position for a time; Thaddeus Stevens advanced the "conquered province" idea; there was a different theory that secession was illegal, so it was therefore impossible, so while constitutional government had been stopped, it could be kick-started by getting loyal officers into power. Eventually, the difference in theories became a factor in a power struggle of Reconstruction: if the Southern States were still in the Union, then Lincoln had the executive's power / duty to suppress rebellion and issue pardons, and could set the terms of getting civil government moving again; but if instead they were out of the Union, then Congress had the power to admit states and so Congress could set the terms of (re)admittance. (This is discussed in McPherson's The Battle Cry of Freedom, ch. 23, sec. 2.)
Lincoln had issued a proclamation on 8 December 1863, offering pardon and amnesty to most all Confederates who took an oath of allegiance to the United States and to all proclamations concerning slavery (read: Emancipation Proclamation). If more than 10% of the voters in a state took the oath, then they could form a state government that the President would recognize ... but as the President wrote in the proclamation, he couldn't make Congress agree and, say, seat their Congressmen. Or, as it turns out, count their Electoral College votes.
Moore, Frederick W. "Representation in the National Congress from the Seceding States, 1861-65." The American Historical Review, vol. 2, no. 3, 1897, pp. 461–471. JSTOR, JSTOR, www.jstor.org/stable/1833400. This article covers Congressional seating of representatives and senators, but to explain the background, it discusses events in two "former" states, and I've looked up Tennessee.
By 1864, large parts of Arkansas, Tennessee, and Louisiana had been recaptured by the Union.
In Louisiana, the U. S. military government had been slow getting off their duffs and reconstructing the state, and a few people's attempt to hold an election in 1863 was nugatory. In 1864, they elected state officers and held a state constitutional convention. In September, the new constitution was ratified, and a new state legislature and representatives were elected. In October, the legislature elected senators; in November, it chose electors.
In Tennessee, local officers were elected in March 1864. A convention met in September 1864 to establish rules for voting for president, including a required oath that a McClellan voter would find hard to swear, and the places controlled by the U. S. government so voted in November (*) A convention for amending the state constitution didn't meet until January 1865, and a full state government wasn't elected until March 1865 (*).
In Arkansas, they started earlier. They had a convention on 4 January 1864 and organized a government. Congressmen attempted to enter Congress that year but were rebuffed. Elections for state and county officials were held on 14-16 March. (Cowen, Ruth Caroline. “Reorganization of Federal Arkansas, 1862-1865.” The Arkansas Historical Quarterly, vol. 18, no. 2, 1959, pp. 32–57. JSTOR, JSTOR, www.jstor.org/stable/40019234.) Sen. Trumball and a couple of others said in a Committee of the Whole debate that he thought that Arkansas, Tennessee, and Louisiana had chosen electors (Congressional Globe, 1 Feb. 1865, p. 534). But Sen. Pomeroy said later that they didn't vote, the ones rebuffed "distinctly understood that states not represented in either branch of Congress would have no right to vote at the presidential election. They returned to Arkansas and so reported, and they never had any election; there are no votes here from that State." (p. 555)
(2) Why? The joint resolution, H. R. 126, as eventually adopted, was
Whereas the inhabitants and local authorities of the States of Virginia, North Carolina, South Carolina, Georgia, Florida, Alabama, Mississippi, Louisiana, Texas, Arkansas, and Tennessee rebelled against the Government of the United States, and were in such condition on the 8th day of November, 1864, that no valid election of electors for President and Vice-President of the United States, according to the Constitution and laws thereof, was held therein on said day: therefore ---
Be it resolved, by the Senate and House of Representatives of the United States of America, in Congress assembled, That the States mentioned in the preamble to this joint resolution are not entitled to representation in the electoral college for the choice of President and Vice-President of the United States for the term commencing on the 4th day of March, 1864, and no electoral votes shall be received or counted from said States, concerning the choice of President and Vice-President for said term of office.
Sen. Ten Eyck offered an amendment to allow Louisiana. In debate, Sen. Trumball (p. 535) pointed out that Louisiana's votes, if counted, wouldn't matter in the slightest. Just counting the non-Confederate states, Lincoln got 212 votes, McClellan got 21. I don't have the apportionment after the 1860 census, but in the 1860 election, the seceded states had 88 electoral college votes, so even if all Confederate states had been counted for McClellan, Lincoln would have still won easily. Trumball pointed out that, in the 1856 election, a question had arisen about Wisconsin's electors; what they did then was to announce the votes both with and without them, and that Buchanan won regardless.
He also pointed out that, if Congress allowed Louisiana's votes to be counted, it would be a statement that they had been reconstructed, and so they would be fully back in the Union, including for Congressmen. He said the resolution above had been amended to avoid the question, and Ten Eyck later confirmed that that was the intent of the Judiciary Committee; Sen. Cowan said specifically that they could not afford a collision at this time between Congress and the President, especially when it wouldn't affect the results. Trumbull also pointed out that Lincoln's proclamation of rebellion, based on a law making it a felony to have contact with a seceded state without licence, had included Louisiana (and the rest), and the law and the proclamation had never been amended. There was debate on reconstruction theories, what the turnout might have been in Louisiana, whether Congress had the power to rule out electoral votes from a state (rebuttal: the "necessary and proper" clause of the Constitution) (rebuttal: so if South Carolina had decided to send up electoral votes, they'd be constrained to count them?), whether the military had interfered in the election by relaxing or imposing restrictions, how there were no procedures for how the convention of the two houses (required by the Constitution to count the votes) was supposed to decide anything, whether it was a retroactive law, Sen. Wade talked about how the Wade-Davis bill for reconstruction had been pocket-vetoed (p. 559), ... I'm glad the House proceedings just said that it was passed.
I would add what I think is an interesting bit on counting the votes. It was mentioned (p. 549) that in 1857, Wisconsin's votes had been called in question. (p. 534, p. 558: There had been a massive snowstorm, and the electors hadn't arrived in the capital on the day required by law for the vote, and the Constitution requires it to be the same day throughout the states. They voted on the next day.) A senator objected to the counting. The President Pro Tem, Sen. Mason, refused to recognize him, but eventually the Speaker of the House beside him did, and various people rose to speak, and eventually Mason led a chunk of the senate out, "indignant" at the thought of the joint convention deciding anything (they complained about "states rights"!). They eventually came to the compromise I mentioned above.
After all of that too-verbose citation, I should summarize. Every senator and every representative has his own reasons, but I think the main reasons tended to be,
So they generally did the minimum that they saw necessary, trying to avoid general principles or too-precise statements. I think Howe was correct (p. 594-5) in saying that 4 days of debate was not about what they would do, but for having well-worded reasons, because restrictions on a presidential vote were so extraordinary.
(3) How? Well, that was part of the debate. When some Senators pointed out that the Constitution just said that Congress was to meet, and the President of the Senate was to count ballots, that implied no power to rule on them. Other senators said it was absurd: the Constitution provided that ballots could be only from states, and only in certain numbers, and the candidates needed some qualifications (age, citizenship), so there would have to be some way to determine the matter.
They did eventually pass the joint resolution.