I have a battery with a protection circuit module where 1 wire from the JST connector has broken off.
How dangerous is it to solder the wires back on the protection circuit module?
It should be fine, peel the tape away and expose the pad, and solder to it. As long as you are not applying insane heat for a long time it will be fine.
It's funny, I actually just had a battery tonight where one wire fell off. Pulled the tape off, and re soldered, then re taped. No issues what so ever!