I wanted to know whether or not K.K. Song can be played while DJ KK is doing his thing. I have stood around his preformances and not heard him play the song.
I can't say for sure in New Leaf, but in most Animal Crossing games, this is a secret song you have to request for him to play.
If there is a way to request a specific song from him, you will have to name it for him to play it for you. Otherwise he will play one of his non-hidden tracks.
I can't say I've seen it for myself, since I don't bother going to DJ K.K. performances. However, K.K. Song is not a hidden song in Animal Crossing: New Leaf, so...
There's no reason why it wouldn't be.
The songs DJ K.K. plays on non-Saturday nights are randomly selected from his main playlist and your town tune. Presumably, this doesn't include the request-only songs, but since K.K. Song isn't one, it would be unusual if that particular song was left out of the list. Since no particular note has been made of this occurring, it seems safe to say that K.K. Song does not receive any special treatment.
I just saw this happen and this question was one of the top Google results I got when I looked to confirm what I saw. I, too, have no proof, but DJ KK played KK Song during his routine with the album cover on the screen in the back. It surprised me to hear Totaka's song and I had to Google to see why DJ KK was playing it.