A.C.
I think this is a decision you have already put a lot of thought into. At this point I wonder if it's truly her decision or the other two girls telling her she is unladylike (!) and that she somehow isn't "looking right" by not shaving. She is only six, and since you said it's not a major deal to her, then I wouldn't let her do it just yet. Just my opinion. I would also be wary of the two daughters of the girlfriend. That they take it upon themselves, without adult permission to shave another childs legs. Big red flag there.
Anyway, sounds like you have your daughter's best interests at heart. Bless your heart for all you do as a single mom. Keep up the good work and stay strong.
A.