This time, I used 制, a word I had learned a while back in the context of 奴隷制度. Basically, 制 can be used to mean "system," as in 会員制, or club system. You tend to see that a lot at karaoke.
Anyhow, I asked the clerk at the Softbank shop if it had a 予約制, or in other words, if appointments were possible. I'm not sure if this is technically correct, but you can see from these ALC examples, there is a precedent.