SPOILER:
Both of them are at least somewhat aware that Edna wants to make this a VERY BIG DEAL. Also, it is clear to Ellery that Wes isn't going to tell Edna that this is not a legal thing, and so it has to be kept rather discreet. No, Wes would not be in any trouble if he does not file the marriage license - however, if someone thinks he did, or makes a complaint that he did, then that in itself can cause problems, and outside of the intimate circle of co-workers, acquaintances, and Wes's sphere of influence, any publicity of Ennis and Ellery living together in a "marital" situation can still be politically risky, for Wes and for Mel, and it could easily lead to more than getting their windows broken or their house spray-painted.