I get where you're coming from OP in relation to your MIL's "tactics". I cannot stand people who do something knowingly and willingly to piss you off/get their own way and when you challenge it, YOU are then made out to be the unreasonable one. This is passive-aggressive manipulation at its finest and I don't blame you for being angry about it. Someone in my family could give a master class in this kind of shit and I have nothing to do with them anymore.
I'm getting from your posts that your MIL is aware that you didn't like the name and that making it a middle name was a compromise on your part. If this is the case, then if it were me I'd do something like this:
Next time she uses the name, remind your MIL in a friendly, calm, light way that your DD goes by her first name only and that the other name was never intended to be double-barrelled with it, so to provide consistency for your DD as she is growing up and to avoid confusion - to stick to the first name. If your DD decides later in life she wants to use the other name that's her call - but at this point and for the foreseeable - it is the first name only. Your MIL might have a tantrum- let her. You maintain calm, keep it friendly, no anger - but don't pander/engage with any tantrums or whatever. Just make the point then move on - change the subject, leave the room, go make a cuppa, go for a walk, leave - whatever to just remove any oxygen or audience from any tantrum.
Assuming she ignores this and does it again, you then get firmer. You've tried the nice approach - now you have somewhere to go with it. You sit MIL down and have a conversation. Again, keep it calm, firm but no anger - just light tone but matter of fact - this is how it is - not up for negotiation type of tone.
You say something like "MIL, you keep referring to DD as xxx-xxx (double name)... when we have repeatedly pointed out that her name is not double-barrelled and we have asked that you just stick to her first name. You know that we don't actually like the name xxxxx, but we appreciated that it meant a lot to you and so out of respect to you we gave DD the name as a middle name. That was not, however, an invitation for you to use it to create your own double-barrelled version of her name. Her given name is xxxx (first name) - and we want her to be referred to as that. We compromised by including xxxxx (name you don't like) as a middle name however, if you cannot also compromise and meet us half way, then we will have xxxx name removed from DD's birth certificate and it will no longer be part of her name at all. We only included it out of respect for you, but if you cannot afford us the same respect then it seems pointless to keep a name we don't even like when it's just going to be used as some sort of game. Please, respect our decision and our wishes regarding DD's name. Her given name is XXXX (first name) - it's a lovely name that we chose for OUR beautiful daughter - and that is how she is to be known".
Again - don't linger over it... don't pander to any tantrums... end the conversation calmly and nicely (eg. "okay - now that's sorted... I'm making a cuppa - want one?") - but make it clear it's not up for debate/argument.
If she continues to do it after that - then follow through. Change the name. No discussion - no second chances - change it. Then inform MIL that xxxx name is no longer DD's middle name so she can stick to her first name as you previously asked. She can't say it's unfair - you have given her more than fair warning and opportunity to respect your wishes.
If all hell breaks loose then I'd be looking at frequency of visits for a while... but don't pander to tantrums and manipulation. I can't be doing with that nonsense - it's tiresome.
I understand you probably want to avoid a drama/massive tantrum etc... but from what you're saying, there's going to be a blow up of some kind - someone is going to snap eventually. Better you deal with it calmly and firmly - if she wants to throw a fit then let her crack on but don't give it oxygen or an audience. Leave her to it until she decides what is important to her - a stupid name or a good relationship with your DD.