So much good advice.
The OP wants to introduce GF to the family.
The DB is in a difficult situation, he want everything to run smoothly.
The DM is an antiquated dinosaur who needs to get a grip.
Someone up thread suggested send out a Xmas card with photo to family breaking the news, excellent idea, very clear, no ambiguity.
Don't put DB on the spot, on the day, say you and GF will attend ceremony, but don't need to be included afterwards if it will make life difficult for bride and groom, it's their day, not yours. Take them somewhere special afterwards.
Reign in your DM. It's between your and her, don't drag bride, groom, and GF into her nonsense. Really, gays have civil partnerships, adoption, surrogacy and now actual weddings, see Elton John and David Furnish, and why not, because your DM doesn't like it?... If it's acceptable for your brother, it's acceptable for you.
Your DM is a hypocrite, call her out on it. Good luck