A person is a person. A person is influenced by religion, but the bottle neck is at the person level. Sure the church can decide if you can use the actual church building if they get wind of it before hand, but signing that note is on the officiators level. You did less misinterpreting, and more misunderstanding where the responsibility of signing a wedding license lies.
A church can later confound the officiating individual's responsibility by revoking their good standing if they preform a non-approved union, but that's after the fact. Anything signed before good standing was revoked will stand in a court of law, else there would be a lot of annulled marriages from Bishops that were excommunicated. Believe me when I say it's not complicated to get another officiating organization to issue you a new one.
Try to grasp the concept I'm throwing out here, because I don't hear anyone mentioning it at all. The officiator, whether that be priest, bishop, father, reverend, captain, judge, or whatever else has rights too. A judge, far less so given that his/her entire salary is Tax dollars.
We can play slippery slope, and it's probably valid. But let's call a spade a spade. If you get free money from the government, you are subsidized. Given that marriage IS a legal union, and your religion offers marriage, you could argue that signing a marriage document is a "contracted" arrangement, therefore the government is paying you to preform. If you don't preform your obligations, the government is paying you to discriminate. Therefore, to get the government OUT of discriminating based on sexual orientation, no more free money.
This is all the more reason to move religions OUT of the marriage business, and give them sealings, or something of the sort. Either that, or make marriage and weddings not a legal union...