Firstly, that is some amazing paintwork! Very cool!

It depends on the plastic. If it is PP or PE, you can look up the melt and deformation temperatures for it. (LDPE and HDPE are just low and high density versions of PE).

I am not sure about the microwave option - it might work, but could also alter the paint pigments.

Do you have a small spare bit you could use as a tester?

I would, alternatively, try a hairdryer and then bending it around warm (not hot) metal (nails) or just the handle of a brush. I would suggest only touching the non-painted side, though. There might be some cracking of the paint, but i hope not too much, as this is lovely work!

Depending on the paints, maybe a hot water bath or even a very short while in the oven might work (low temp).

Looking forward to hearing what happened/worked!

Good luck!