Surgery may not be billed incident-to which I did put in my original response, which Sharon noticed.
Regardless of participation status, regardless of who the primary surgeon was:
1) Your PA name goes as the rendering provider (box 24J or electronic equivalent)
2) Modifier -AS on CPT that allows an assist.
3) I imagine you bill as a group, and the group name/TID, etc goes on the claim as well (box 25 & 33 or electronic equivalent)
4) The supervising provider is you practice's physician that is supervising the PA. (box 17 with DQ to indicate supervising physician & box 17b)
If there are still some insurances that do not credential PAs, they may have a policy to bill under the supervising physician. In absence of a written policy by a carrier to do so, the claim should be billed with PA name as indicated above.
Intentionally changing the name of the provider for the purpose to get a claim paid that would not otherwise be paid is not something you want to be accused of. Whenever hiring a new clinician, the day the contract is signed (and typically well before start date), I start the insurance credentialing process. If you expect to be paid for a clinician's services, they need to be participating (unless insurance has out of network benefits and pt is informed they are using an out of network provider.)