My MIL used to do this. One year, DH picked out a camp and suggested it to BM, who agreed (which was not easy because BM is contenious). MIL learns about the camp and then calls DH and says, "the boys say they don't want to go to the camp you picked, so I went and signed them up for this other camp. Here's the number to call to pay the difference in fees." This was before DH and I were married...I lost my mind on him, because he just went along with it. Then BM tried to pick a fight with him because she did not agree to the camp MIL put them in (she really didn't care, but anything to pick a fight). Not sure why these grandmothers can't understand that parents should make decisions for their children, not grandparents. My MIL will now make plans directly with SSs without including DH (BIL will do the same). We just ignore the whole thing now, because DH has asked her not to and she just bulldozes him. Then she wonders why I rarely want to see her and why she doesn't see my kids that often. That's the only thing I can do, hold the boundary that I don't have to see her and refuse to go along with anything that will disrupt my plans. 

It's your DH's time. I would be relieved she's out of your hair for a few hours each evening if you're being put in the position of taking care of his kid. If I had it to do over, I would charge for child care time. 

stink on the shit that MIL is. If I was BM.

Just to rub MIL's and DH's nose in it all.

Stand  your ground on not accomodating MIL's over stepping. She can pay for camp and transport the SKid since she is the one overstepping. Stand on no, and help DH stand on no as well.

MIL will figure out transportation of SD.  You didn't enroll her. Not your problem.  

she can pay for and transport to and fro for the entire duration. if it comes to fall on you, "No." is an entire sentence. 

Fact is nobody gets to be a dictator and makes executive decisions regarding your time and presence. The more you let this go, the worser it will be. 
 

i told my husband many yrs ago that it was frustrating even with inlaws giving last minute notification like day before we were flying to my country to visit my dad with fabric so we could get it tailor made into a dress and be matching with family at the wedding barely days after we arrived back home.

other times hubby and sd's dictating our time which involved my time, privacy and presence. I ended that many yrs ago because it is disrespectful. I don't answer to anyone else

Unless the MIL and bio mom and your husband are transporting AND paying for said camp then the answer is NO 

But if you want the kid out of your hair so you can have a quiet house then maybe work with them a bit but make your husband compensate you for accommodating the request

I guess this is a question of if you want this to be the hill to die on?

If not, let it go. You don't have to do anything for it and once a week she is out of your hair for the evening.

If you want to make a point of it, call the dance studio, let them know that it wasn't a parent or guardian that signed her up. She does not have permission to go to class. Tell MIL that you have disenrolled her and next time ask first. It was rude to just assume she could make decision about a child that isn't hers.