Think I've figured it out.
Before we get into this, let's take into account a couple of things.
1. Togami spent a lot of time in the library, so much so he made a habit of getting rid of anyone who came in.
2. The latest update confirms Chi is very likely male.
What I believe happened is that the culprit killed Chi in the male changing room. The obvious mo there is to pull a switcheroo and make it look like only a female could have killed him. What I think happened after is this. Togami found the body, recognized the cord used to hold Chi up. He takes advantage of the situation and immediately places Syo's mark on the wall, setting his plan into motion. He realizes the culprit has to have gone into the library, more importantly the room with those classified files which is his main reason for placing the mark.
Having set up the bait, he gives the culprit an ideal out, all they need to do is to go into that room, take the aforementioned file and they can place the blame on Syo. What they don't realize is that with Togami having been in the library all the time, the only way they could have known the file was there was by having been in the room before, giving them the opportunity to take the cord as well. He uses Naegi(like I mentioned before) to prove the existence of the file itself, and simply waits for the trial for the killer to pull out the file, at which point he can put the screws into them and end it. Like clockwork, fat boy has already told Naegi he has decisive evidence, and the file just happens to go missing.
Well that's my theory at least.