The writer of the song intended it to be archetypal. The northern people accepted it as archetypal and adopted it as their de facto war anthem.
You could cut the ground out from under this argument by calling the song mythology. I would have to concede your point.
Why not just quote Lincoln on why the war was fought?
"As to the policy I "seem to be pursuing" as you say, I have not meant to leave any one in doubt.
I would save the Union. I would save it the shortest way under the Constitution. The sooner the national authority can be restored; the nearer the Union will be "the Union as it was." If there be those who would not save the Union, unless they could at the same time save slavery, I do not agree with them. If there be those who would not save the Union unless they could at the same time destroy slavery, I do not agree with them. My paramount object in this struggle is to save the Union, and is not either to save or to destroy slavery. If I could save the Union without freeing any slave I would do it, and if I could save it by freeing all the slaves I would do it; and if I could save it by freeing some and leaving others alone I would also do that. What I do about slavery, and the colored race, I do because I believe it helps to save the Union; and what I forbear, I forbear because I do not believe it would help to save the Union. I shall do less whenever I shall believe what I am doing hurts the cause, and I shall do more whenever I shall believe doing more will help the cause. I shall try to correct errors when shown to be errors; and I shall adopt new views so fast as they shall appear to be true views.
You could cut the ground out from under this argument by calling the song mythology. I would have to concede your point.
I would say it was a song and not a statement of administration policy.