The final sentence of the Gettysburg address seems to contemplate an existential threat to the existence of the United States of America:
It is rather for us to be here dedicated to the great task remaining before us—that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion—that we here highly resolve that these dead shall not have died in vain—that this nation, under God, shall have a new birth of freedom—and that government of the people, by the people, for the people, shall not perish from the earth.
So, my reading of this is that the government Lincoln refers to here is the US of A, of course.
So suppose that the CSA defeats the USA in the civil war-- or at least fights to a peace. That would leave the CSA existing as a country, with its constitution largely a copy of the United States Constitution, except that it institutionalizes slavery.
So the CSA, as a representative democracy, is arguably a government "of, by, and for the people" insomuch as the United States was originally founded as one, only enfranchising free white males as voters. Neither countries are monarchies, for example.
Is Lincoln saying that the CSA, by instituting slavery, cannot be a democracy? How could the United States be considered to be such a government, with its three-fifths compromise, and free- and slaveholding-states compromise?
You're over thinking things a bit. What Lincoln is referencing is a bit more basic that that, namely the idea that the war itself was about the principles of republican government. I expand upon this more here both in terms of Lincoln and wider Union sentiment, but to briefly summarize, it comes down to the idea that there was an election, which everyone participated in, and when it doesn't go the way part of the country prefered, they decided to leave rather than respect the results. Thus in the minds of many in the North, to allow the breakaway states to do so would essentially invalidate the idea of elected government, and so they believed themselves to be fighting for the very survival of that idea, and that if they failed, it would be a failure of the very concept of, well, "government of the people, by the people, for the people".
"Of the people, by the people, and for the people" is a rewording of a quote from Daniel Webster's Second Reply To Hayne, a speech given on the Senate floor in 1830 that is often considered the first important refutation of the "State's Rights Doctrine". The speech contains the line:
"It is, Sir, the people's Constitution, the people's government, made for the people, made by the people, and answerable to the people."
The speech was very well-known in the 19th century, and reprinted widely. Joel Parker wrote in 1853 that "every American knows" the speech "by heart". The Granite Monthly wrote in 1878 that it was "the grandest speech of all time".
It was a deliberate wording by Lincoln to insinuate to the American people that his view was the nationalist view, the Webster view of the Constitution, and a rejection of the Hayne/Calhoun view and the "States Rights Doctrine" of the Constitution. The American people would have readily understood the inference of the line. So, yes, it was definitely a deliberate rejection of the Confederate point of view of the Constitution.
Upon the publication of Webster's original speech in 1830, even James Madison agreed, writing to Webster that he approved of "its overwhelming effect on the nullifying doctrine of S. Carolina".
A couple months earlier, Hayne had written to Madison to get his opinion on the debate, to which Madison responded he believed Hayne was wrong. Hayne replied that he rejected Madison's opinion, promising to elaborate on why he believed so later. In Madison's collected papers, he seemed to take a dim view of the exchange, with the note on Hayne's promise:
"The promised communication has not been made May 1835."
Anyway, Lincoln chose the words deliberately, to support the decades-old nationalist view of the Constitution, paraphrasing the words of probably the most famous defense of that view given before the Civil War.