So your life after PIP as you have figured out is not nice. Either you get fired (with severance), or you are kept but are still tagged as a low perforer which affects your next appraisal (s). Not to mention your peers are are laughing at you if you care about social niceties.
So use this PIP time to do some leetcode and get hired. Trust me. There are 1000+ companies still hiring. MNCs have a hiring freeze but join a startup (culture is same since you're already in Amazon). Also it's easier for you to get shortlisted due to FAANG tag and easier yo get in (I'm guessing you did some level of leetcode before to get in)
Worst case is you finish your project in PIP. As I've mentioned, this cannot be good for your future.
So do leetcode and prep. PIP exists so that people can leave voluntarily. If Amazon fires you they will have to pay you severence. This is them trying to avoid it.
Happy leetcoding