The pcb fragment has nothing to do with the battery clip, it has broken off from the other half of a pcb and it looks like the one on the right in your photo. Should be an easy job to repair if you're any good with a soldering iron. Just glue the board back together and bridge the tracks. If you're not, then I'm sure that there is someone you know or a luthier could fix it. If you end up not being able to get it fixed, if you could get the broken bits to me, I could make you a new pcb.