Halfway through my resignation period my manager hit me with a PIP.
This is a red flag in any case, and highly unusual. Not much point in improving the performance of an employee when they are already leaving. Unless you have an unusual notice period when they want to cut it down unethically using clauses of PIP. Or if the manager is trying to sabotage your career record (Employee X got a PIP is on the record for you)
I refused to recognize the PIP and accelerated my last day
Is the HR aware of any of the above steps so far? If not, they need to be involved.
He then called me to tell me I was under investigation for downloading company software to USB.
This is a very pointed and highly serious charge. The type of charge that is cause for being fired at most places, and gets mentioned in your work records because it would fall into category of espionage. Think it through completely - did anyone else use your system for downloading such a thing? May be you forgot to lock your screen? Be sure that you are on the right side before proceeding further. Be doubly sure this is not a trap - you later mention change in balance of power, could there have been motivated physical sabotage involving your account? Are you being set up?
He would forget about it if I just admitted it.
Once you admit, his current feelings that he states do not matter, he/the company has evidence you are guilty. This is him making you shoot your own hand. Check with a lawyer before replying in any manner.
This is all an attempt to derail and sabotage my move to the new segment where it would shift the balance of power. Do I have a case for a lawsuit?
Based on your description so far, this seems correct. But only a lawyer can answer if this is a lawsuit case or not based on your jurisdiction. Your manager has definitely been harassing you, but only a lawyer can tell which laws he has broken.
So far, mental trauma, defamation (if investigation is leaked) come to mind.
TLDR: If I was in your shoes, I would immediately run to a good lawyer.