The thing here is first of all is that house rules are house rules and should be stuck to by everyone.
Enforcing these house rules should not be your sole responsibility. DH needs to actually get up and reiterate to his DD that these are his rules too and not let you he painted into some type of wicked step mother role.
It is not at all uncommon for teens to have these sort of remarks and make these sorts of ultimatums at this age. DD is using the step parent angle l because that is the available ammunition, she would still say/do similar things even if this were not the case.
The DD’s behaviour is not about you, it’s about being a teen thinking she is on the verge of adulthood who doesn’t want to be told what to do by you or anyone else. She has to be shown and taught that regardless of how adult she thinks she is, and that there are rules that we all have to abide by and there still will be rules even when she is an adult...that’s just life!
DD also has to learn that there are consequences to her actions, it’s now up to your DH to decide what those consequences are. He might like to explain to her that while he loves and cares for her very much he will not tolerate someone (even his DD) not respecting the house rules, and that as his wife you pay for this house too and that it also your right to enforce those rules too. He might also like to add that now DD is old enough to chose who she would like to live with she is free to make that choice. He will always be sad to see he go of course should she choose to live with her mother instead, but that she is always free to return home with him/you but will always be expected to abide by the rules of the house. It’s up to her.
If he has the conversation above it removes her ammunition, there will be other battles of course and she may still choose to stay with her mother (though I’m sure her mother has rules too) but this should hopefully nip in the bud regular threats of leaving and her going back and forth between the two houses. If DD is allowed to think she has your DH over a barrel on this matter, than she is going to keep on using it and he is going to be tying himself up in knots to keep her from leaving every time she doesn’t like something.