Well, I don't know if my advice will help, but...
Compromise is definitely something she will need to understand once you're married. If this is your dream, she needs to realise it, just for you. It's a once-in-a-lifetime chance.
-Do the traditional groom/bride dance.
-Explain to Quadrette how important the mother/groom dance is to you, and how important it is to your mother. I don't think either of you want to estrange your side of the family. A wedding is a family affair, isn't it?
-If Quadrette is upset about the fact that she does not have a father, ask if she would feel comfortable dancing with YOUR father, as he will be her father-in-law in the future, anyway.
-If she is not comfortable with your father, ask if she would be okay with her brother or male relative.
Explain to Quadrette how important this day is to you. It is a one-day affair, and it wouldn't hurt her to go along with it. However, you better be ready to do her a big favour, as well.