that's a rough choice on a tight timeline
if you're open to suggestions: take the severance package
anecdotally (including my own anecdatum) a PIP is just a step in building a case for constructive dismissal, especially if your workplace is a factor in your burnout. do you really want to have to work even harder for a chance at staying there?
take some time to recover from burnout (even if it means having less funds for a bit), then try and find something to do for employment thay doesn't suck so hard
myself, I chose disappearing in the bureaucracy of a multinational enterprise for a reduced salary and title. getting away with only a few hours of work most days, and spending the rest of the time living my life