Can I use cat 6 to power a wired doorbell (not elite)?

I installed Cat 6 in preparation for using POE with the elite doorbell. I was not made aware that the elite is NOT a flush mount and I don't have the ability to install it in the exterior wall. Frustrating. So, I'm now looking at using one of the Wired pro doorbells as an alternative. Can I use that Cat 6 wire to power the doorbell with a transformer in my network room? If so, which transformer?