"with an invoice showing the amount billed for rental, nothing about it being a donation."

Because it was Not a Donation. It was Rent Payment. Then, your client gave that right to the Church, so there is no Donation to the venue; there is Payment to venue as rent.

What you have here is Church Fundraiser activity. The Church didn't give them anything showing that rental amount, but your client has proof of the value they gave, as a PrePaid Venue.

The venue is not a party to any of that part between your client and their Church.