I like the thought process here....
So do I. Instead of a “wedding”, you could have a “Celebration of Holy Matrimony”. That distinction could give the cake bakers a way around the onslaught being unleashed on them.
I suppose that priests and ministers would have to forgo signing the legal paperwork. A “Certificate of Union in Holy Matrimony” would be a separate document, like a baptismal certificate, issued by the church. We need to find more succinct terminology for all this, but I guess that is where it will have to go.