I'm fairly sure that the very first time you talk to him, he will automatically play something. However, I can't completely confirm this at the moment. The following method may work the first time, but as I've already spoken to him and had my first encounter, I can't try it.
What I can confirm is this:
In order to request music from K.K., you need to back out of the first conversation with him. When he asks if you want him to play the first time you talk to him in an evening, decline, or else he will automatically pick a song to play at random. I tried this tonight because I didn't want to be given another random song like last week, and it worked.
When you talk to him a subsequent time, he will give you 3 options:
- Request a song ("I know a song!")
- He'll suggest one based on a certain mood ("Here's my mood...")
- He'll choose randomly ("I'll let you pick")
The last option you'll have is to back out, which is "Never mind." You can keep on requesting songs for the duration of his stay on Saturdays, but you'll only receive a copy of the song for the first song of the night.