First, I'll say it is totally and completely up to the leadership of the church to set policy. Whether that policy is to let non-members play or to specifically exclude non-members. Set the policy and make certain everyone is aware of it. It has to come from the leadership and it has to be followed by all.
Now, here's my situation(s)
For years I played bass at churches where I was not a member. I would have LOVED to have played at my church, but at the time we were stuck in a tradition of "music = piano &/or organ". Okay, fine. So I played where I was welcomed to play. Sometimes I was paid. Sometimes I wasn't. Even when I was paid, the money typically went back into the collection plate or to upgrade equipment used exclusively for that church (strings, practice amp). I was not a member and knew I would not be a member, because I already had a church home that I was very happy with. I provided them a service. They provided me an opportunity to hone my craft.
Eventually my church added an early morning service. I have been in charge of the music for that service. I brought in a keyboard player from one of the other churches I used to play at. I knew he had no intention of joining our church, but since our service was held early, he could play at our service and then make it to his own church in time for his service. I doubt I would have hired someone who had NO church home. I definitely would not have hired anyone who did not possess and display the spirit of Christ.
That keyboard player took a job in another state, so I hired another keyboard player who leads an ad hoc group that I play for. Eventually he hired me to play at his church. So now every Sunday, he starts at my church in the morning, then travels cross town to play at his own church. On two or three Sundays a month, I start at my church and then drive cross town to his church. I don't expect that he will ever join my church and I know that I will never join his church, but we have become great friends and musical collaborators and we both provide dedicated service to both congregations. I see nothing wrong with that.
If, however, his church decided that they wanted to institute a policy that you had to be a member to be a musician, I would gracefully step aside.
Peace,
James