They can accomplish their goal with a private security force, just without legal authority of arrest and imprisonment. Private security can detain until legal authorities arrive. That will work just fine and not violate the constitution.
another question worth asking: who's authority would this police force be under? The Brairwood Church session? ("session" == elder board, in presbyterianese.) Do they want, and can they handle that responsibility? What happens if and when they get sued over the behaviour of "their police"?
To my understanding, if they’re bestowed full police authority, there would in most case be sovereign immunity, with the exception of instances of abuse in which case they could be successfully sued.
I don’t want any church of any kind to have sovereign immunity in this country, ever.