Is a Non-ERISA 403(b) church plan subject to ACP testing?

No, it is not subject to ACP testing.

A somewhat harder question is whether a non-ERISA church plan is exempt from the excise tax on excess aggregate contributions. I believe the answer is yes, that it ought to be exempt from that, but the last time I researched the point years ago it was not really that clear.

Why would it be subject to the excise tax if not subject to ACP testing?

I think it is exempt from the excise tax (which is the sensible conclusion), but finding a citation to support that exemption is difficult. Like I said, I've not looked into the issue for a long time.