Shelby Foote's 3-volume history --The Civil War: A Narrative, volume 3 (obviously!) page 958.
" . . .(Lincoln) proposed closing the interview by having 'the musicians play a particular tune which I shall name. I have always thought Dixie one of the best tunes I ever heard. Our adversaries over the way attempted to appropriate it, but I insisted yesterday that we fairly captured it. I presented the question to the Attorney General and he gave it to me as his legal opinion that it is now our lawful prize. I now request the band to favor me with its performance.'"
on the previous page the date is given as April 10 (1865), the day after Lee's surrender.
Nothing in there about honoring the Southern soldiers, the words "capture" and "lawful prize" stick out. My copy of Sandburg is an abridged version and doesn't have that story in it; hard to imagine it being dated the day BEFORE Appomattox.