To me they are different names. I used to know a Peirs - pronounced Peerz, whereas Pierce is like Pierce Brosnan or to pierce something with a skewer IYKWIM.
I agree with Mummy Tummy - only because I met a Peirs (pronounced Peerz) - although I think it was spelt Piers.

Overall though, sweetie - its your name & your baby, so you do whatever you think