Let's see - this thought is driven by your thread title ("Emotional Burn Out"), the statement that supporting characters need to put pressure on the murderer, and the "I guess he'd just kill himself" reaction of your murderer player. I'll assume you're interested in tragedy as one possible - maybe even likely - outcome of the game, but not the only one. That is, I'm assuming that this isn't a game plainly and only about flavors of tragedy. I mean, if it is, that's still PLENTY of creative room, but I think the constraints on that would be a bit different, and what I'm about to say may not apply.
So with that out of the way ... while it is entirely sensible that the supporting characters need to put pressure on the murderer, the murderer also needs SOMEONE to reach out to for solace and support - with some chance to authentically get it (did I just say a murderer needs solace and support? Good Ghu, I think I did). Given the situation, "authentically" is going to be hard to come by, but it shouldn't be impossible (again, unless pure tragedy). While that solace and support MIGHT (in some sense, in some rare circumstances) eventually come from the ghost, I imagine it'd take a lot of fiction and fictional events before that could happen. If the supporting characters are also entirely, um, aggression-oriented from the get-go, the murderer as a fictional character ends up with no where to turn, and the murderer-player might easily give up on their ability to do anything interesting with the situation.
It's actually neat that the game manages to drive the murderer-player into that state, but what I'm sensing in this post is that the murderer (and the murderer-player) ended up being TOO isolated. I'm not sure if the design needs something mechanical to help avoid that (incentivize supporting-player(s) to move ghost-circle characters into shared-zone?), if something in the language could clear it up (make players label a/some supporting characters* "friend of [name_of_murderer]"?), or if it's entirely an unavoidable possible outcome (maybe enhanced in a three-person game where the supporting-player really applies the pressure?)
But that's where I see to look - hope it's helpful.
*I note that the current draft includes no examples, and the language on creating the social circles says NOTHING about categories or types of people/relationships to consider including, beyond "involvement in crisis/murder".