I "get" you completely..
.. I don't feel you are being "over sensitive".
I can see the logic of letting others help out, and your dc having experience of more than just you, but in the case you describe, your MIL may as well be a stranger off the street, as far as your little one is concerned.
What's worse, is that your MIL isn't even staying around.. so as you say, what's the point of getting your baby "used to her", if in a couple of weeks, it's "bye-bye" til possibly next Christmas... and she will still be like a total stranger.
If someone took my distressed baby from me in order to try to "calm the dc down", I'd be probably be a weird mixture of strained relief and totally stressed at my ways being forcibly removed from me.
It doesn't make much of a difference if your MIL is being intentionally bull headed, or if she's innocently trying her best to give you a break.
The outcome is that your dc is getting even more upset, which is then winding you up further, coz you know that if you'd been able to get to your dc from the off, it'd have been quicker and much less stress to get everything quietened down, and therefore everyone else in the house would be able to resume sleep more quickly too.
I really can't say the answer as to how you can get your dc to be more open to others that are happy to take over from you in times of need.
I would be just accepting dc change for the moment... dc was fine before, and I guess this is a phase.
The more stress shown to your dc, will be picked up on, and result in this phase lasting even longer.
Your MIL reacting to you in the way that she is, is immature and she is not subtly letter you know that she feels "hurt"!just because you enforced your method over hers.
As is often said on this forum
"your dc - your rules"....
Try to focus on the relief you will feel when your MIL has packed up and gone home...
.... try also to think that (unless you know better), MIL was trying to act for the best out of kindness...
Good luck.... your a better and stronger woman/mother than I, in that you've had your MIL with you for ever so long, and still pretty long to go yet...