I bought some hand warmers from Decathlon store and activate these in the pouch I have my batteries in to keep them warm prior to flying. I have this winter repeatedly had long periods where the aircraft sits on the ground with a "warming up" message and being unable to fly until it is warm enough. Even transporting the P3 in its rucksack in the passenger seat of my car.
My question is; could I put a hand warmer in the battery compartment on my P3S to warm that up prior to putting the battery in, without causing any issues?

Definitely
A lot of pilots actually do that
Your batteries should always be warm (at least 20°C I think; that's what I do)
About the warming up
You should do a cold IMU calibration
It solved it for me
I never get that message again and can fly almost right away after I power my P3P on

I cant say weather or not it would do harm to your drone but if you do it you should still let it hover in front of you for a min or so to allow everything to else heat up.

be careful they don't leak, i believe the have iron filings in them that would wreak havoc on the battery if they got in! Im not sure but i think it trashed one of my p3s batteries that was not very old

Hovering in front  .... never bothered. Once warmed up and Safe to fly shows - I'm away !!

Nigel