I know the mom vs. dad thing is a tangent that the OP didn't bring up, so I'll try to be brief.
While I think that most people can say that dads don't care about hair, I don't think it's fair to make that decision for all dads.
My DH has opinions on how our kids' (we have one girl and one boy) hair should be. He also thinks, and I concur, that it's important what the kids think of their own hair, and they're still little.
I usually style their hair, but DH has attempted to comb DD's long hair and put in ponytails. I have 30+ more years of experience, but for a newbie, he does a good job and I applaud his efforts.
The other day, I suggested to DH that I might give our DS a flat-top haircut but I ran it by him first. He was strongly opposed to the idea. DS looks a lot like DH did as a kid and DH has really bad memories of having a buzz cut because of a traumatic incident at school. When DH was 5, his parents were missionaries. They lived in a place where everyone was one race and DH was not that race. DH was the only child with blond hair. The kids teased him and ganged up on him and dumped house paint is his hair. His parents couldn't get it out and they had to shave his head. Seeing our son in a buzz cut would remind him of this, so out of respect, I'm not going to buzz DS's hair.
I think what OP's FSIL's mom did was a huge boundary violation and disrespectful. I think FBIL and FSIL should tell her they are very upset even if it falls on deaf ears. They have a right and duty to speak up in defense of their boundaries and their DD. And I really don't think they should let her mom see any kids unsupervised because she lied and disrespected them. She has not shown any remorse.
And I agree with the PP who said that the fact that the mom lied about it shows that the mom knows it was wrong and would not have met with their approval. I will add that the fact that the mom did it while FBIL and FSIL were not there also indicates that she knows they would not have approved.
And I wouldn't accept a non-apology from mom either. A huffy "Fine. Whatever. If you're so oversensitive and it's such a big deal, I apologize. Sheesh. Get over it" is not an apology. Unless mom truly showed remorse and I trusted her not to do something like this ever again, I wouldn't let her babysit.