i think one short visit by your partners mother is unlikely to cause significant, lasting detriment. I think making your partner sad, upset and distressed by refusing to allow their mother to meet their child, when this is something that is important to them, could to cause lasting damage to your relationship. Which I think may also be to a new mother’s detriment. No, MIL can’t stay for a few days- but she probably can pop round for a short visit with the understanding of time limitation/giving back the baby if asked etc. I doubt 30 minutes or an hour is going to damage her hugely. Unless, as I say, MIL is truly awful/abusive etc.
And it’s impossible to say whether OP’s mother has done something terrible, mildly annoying, or nothing particularly consequential. I have experience of both ends of this spectrum with 2 family members- both don’t like and avoid their MIL, and would tell a similar story re the reasons but the situations really aren’t the same. So, I don’t know whether I’d be 100% sure that the MIL is definitely awful and the OP’s partner totally in the right, based on what op has said and their partner’s stance on her MIL. Could be either, or somewhere in the middle.
In my examples- one case the MIL is a complete nightmare- cutting, sarcastic, belittles others around her. She is really unpleasant. We’ve all seen it and quite a few been on the receiving end of it- even those she barely knows (the MIL is not a direct relation of mine, but I have experienced her behaviour first hand). I totally understand why she keeps her MIL at (several) arms length. Her husband does recognise it, though I would say has in the past played it down a little. I think it’s better for the children that contact is minimal and always supervised.
The other situation really was a case of personality clash, neither party totally right or wrong, and a series of small issues (about 50/50 in terms of who did what) culminating in a more significant falling out where they were definitely both in the wrong (her MIL possibly slightly more so as she triggered it- though I think either of them could have been the catalyst, it just happened to be her on this occasion- but not malicious intent from either). They are quite different people, and I can see why the DIL in question might find her MIL a bit irritating/hard work and difficult to like- and vice-versa. But, MIL is not a bad person and does care about her family, though she can be quite stuck in her ways. She is quite close with her son. The DIL (my relative) is generally also a nice enough person, good fun, can be thoughtful/generous etc but can be quite certain of herself too. She is also quite reactive, has a tendency to take umbrage very easily, and is somewhat prone to holding grudges. Like many of us, she is also not very good at admitting fault/apologising. Her MIL has apologised for her part, but because she will accept total fault (and grovel/agree to certain things), DIL won’t budge- she won’t see MIL and tries very hard to limit her children’s contact with her (MIL is categorically not toxic/racist/bigotted or a safeguarding concern to children).
This is against her husbands wishes- I don’t mean just won’t visit MIL/allow her in her home- her husband has said that she will make it difficult for him to take them to visit his mum or attend family events without her- e.g. by planning other things if he mentions making plans, makes a fuss/tends to be quite huffy or passive aggressive if he insists on taking them anyway and will create sudden emergencies to try stop them going/result in coming back early. It makes life difficult for him, practically and emotionally, when it comes to his family. I know it has caused significant problems in their relationship, possibly terminally so (my DH is also her DH’s friend) as he feels hurt that she can’t recognise the impact on him, nor that she contributed to the situation with her own actions, and now will not compromise at all. She appears to be using her children to get back at her MIL. If you asked her, she’d tell you MIL was an insufferable, interfering horror who delights in upsetting others and can’t see that she is unreasonable. She genuinely does not seem to recognise that she’s partly responsible for the situation and feels totally justified in her stance. She would also say her DH “doesn’t see it” and makes excuses for his mother. I imagine he probably has, on occasion, but his mother has not been universally in the wrong and he has also done the same with regards to his wife to his mother and wider family and friends…… I think they both did and said some things which they shouldn’t have and ought to be able to find a way to co-exist to some degree.
All I’m saying is that new mums can still be wrong, be unkind/controlling etc- they can do things which have longer-term consequences upon their relationship and those with wider family members. And they are not the sole and final arbiter of everything to do with their baby. Distressing your partner over something like this (especially if it’s just make a point or to upset your MIL) is not ok.