Run the operation as a corporate flight department. You can fly anywhere in the world without a 135 certificate. Pure part 91 operation. Just make sure the paper trail shows the department as being part of the church or corporation.

If that doesn't fly with the church you can run it as an aircraft management company. The church leases or owns the aircraft. You handle scheduling, maintenance, pilots, mechanics, and training for a fee. Again, it is a part 91 operation.

The aircraft lease can be aircraft you own, or someone else owns. It can be a per hour lease or a yearly lease.

Good news. Being a part 91 op you don't need any 6 month checkrides if you are not flying jets or large aircraft. No duty time restrictions. And, no TSA screening for the passengers.

Non-profit or for profit status doesn't matter.