Your insurance company will have their assessor assess the damage and work out a price the repairs. They will then get the repairs done by a repair company that will do the work for that price or...
If they're of the same chemistry and voltage - probably. But if they are too different in size the charge current can be too big for the smaller battery.
The PSP-Go has an internal (nonremovable) battery and comes with everything you need to charge it when needed. Average battery life is ~6-10 hours depending on what you are using it for.