Nope, you're not uptight. Nope, permanent relaxing shouldn't be considered. Nope, SIL shouldn't have any say into Dd's look.
If DH is taking DD to grandma's for weekends, and neither DH, grandma nor SIL know how to care for Dd's hair, then you may need to style it in some sort of way that lasts throughout the weekend - and may start teaching DD how to use a scarf each night to maintain the look.
A bi-racial girl's hair doesn't have to be straightened to look nice - but when it's not taken care of (as in styled some sort of way - even if the curls are just finger combed), it can look quite unkept.
Moms are always going to disagree on when their daughters should be allowed to wear make-up, nail polish, hair coloring, etc. I think that things can be allowed gradually over time. I'm not going to allow a 10 year old to wear make-up because she'll probably rebel if I don't. Just like I'm not gonna allow a 15 year old to drink beer in my house because if I don't, he's gonna go somewhere else and do it.
So, you have choices. You can teach DH, grandma and SIL how to care for DD's type of hair, or you can style it to last for the visits.
Good luck!