Should I replace my battery?

Great work young technician! You’ve done a great job by disassembling this power supply. If this would happen to me I would go for this battery and mount it back with no bothering.

The good practice thou is to replace an damaged/used (for at least 1y) battery. Since your component didn’t blow up then it’s fine, but it could not handle the capacitance and act as faulty.

Replace it if you wish. For further management: if the battery doesn’t feel like PlayDoh/you do not see any internal layers or battery is not puffed or bounded then it is fine. But be careful - power supplies are dangerous as #$%^ - you never know when it will blown up so keep your window open and your throw as far as possible ;)