You should dust off your resume and find a new job when people you work with start physically fighting. Staying in that sort of toxic work environment will not do you any good in the long run. That toxicity is likely to leak out to the rest of your life too.
As for right now, don't get involved and avoid getting hurt. If you really wan to give 100%, perhaps try using some agile practices to make the meetings more about the tasks and less about the people. Kanban and scrum do something called walking the board where the project manager goes over all the projects on the board generally starting on the left and working towards the right. This keeps people engaged by taking the pressure off of them so they don't end up sitting in a 15 minute meeting rehearsing what they're going to say until it's their turn to say it. Perhaps something like that could salvage this situation.
Don't kid yourself though, this is definitely a find a new job situation.
Since this is entirely between clients, sit them down and explain that you'll have to drop them both or work with them entirely separately if they don't get along. Firing your clients isn't ideal, but it's something you have to do sometimes.