Haha, I don't know that he could get up to 180ish bpm just by doing that. Not to diss Sinon's behind or anything.

I know you're kind of just joking with your post, but I think it is interesting to consider this.
Any scenario involving confronting the murderer is pretty dangerous. Let's say he did log out, options then are:
1) Call police on Shino's apartment
2) Go there alone
Both involve the murderer being trapped in her apartment unless he can escape out a window or something. It would potentially turn into a hostage situation.

Given the scenario, the absolute safest option seems to be to defeat Death Gun in game and hope the murderer follows the code they have set. If DG gets knocked out of the tournament, hopefully the killer leaves quietly. If, by fighting Death Gun, Kirito remembers the guy's SAO name or gets him to tell him, then they can look up his address and catch him. Then you would hope to catch any co-conspirators via interrogation/plea bargain or by raiding the guy's phone/computer logs.