Well, welcome to the club! Other noteable groups objecting to their principles being regulated by a government office include Masterpiece Cakeshop of Lakewood, CO, and Memories Pizza of Walker, IN. (For the moment, disregard the likes of Hobby Lobby and Little Sisters of the Poor, as their matters of principle-regulation are less directly relevant.)
I figure there are three-ish main options.
1. People are consistently required to suppress their principles, and do business with groups like the Daily Stormer.
2. People are consistently allowed to exercise their principles, and refuse service to gay weddings.
3. A disaster area of conflicting regulations both for and against the right of various groups to be served by various businesses, conforming to no consistent set of principles but rather to whatever is politically popular and expedient today, and hypocritical to the core.
My money's on 3.
(There's a theoretical possibility they'll actually nail down specific principles and not make it a total mess, but I don't think it's plausible.)