Try any of the local Methodist churches-they seem to be more willing to have non member weddings. Also consider having it in a place other than a church, and use a minister off the secretary of state website. I am registered to do wedding ceremonies also.
We had the same problem. There are actually quite a few, but sometimes the fee is higher than you'd think. We checked downtown Dayton, there are several that will perform ceremonies for non-members: First Lutheran and another on I think 3rd Street across from Square One churches, the First Baptist church on Monument, Christ Methodist by Fairmont h.s. I think Fairhaven Church on Marhsall/Whipp. There is also Normandy Church on W. Alex-Bell. You also might want to look into other venues like: The Old Courthouse, Memorial Hall, Carrillon Park (all through DaytonHistory.org), the Masonic Temple, the Dayton Racquet Club, the Dayton Engineers Club, and the Dayton Art Institute, Polen Farm, Benham's Grove. Again, the fee for non-members at churches are astounding (or at least it was to me!) but we had to go a different route than Catholic at this time.