I've not read the whole thread, just the OP so I'm basing my response on that alone. There's a lot of 'my wife thinks/says XYZ about my mum' but not many clues about what you think is really going on and which one of them is the more 'difficult' personality here. It's also hard to know whether you do unreasonably take your mum's side over your wife's, or whether you are completely fair, measured and rational but your wife won't be happy until you've severed connections altogether, because she sees this as a battle where there can only be one winner.
What I will say, is that regardless of the answer, you do not need your wife's permission to take your daughter to visit your mother. Your wife can dislike your mother all she likes, and maybe she has good reason. But unless contact with your mother is actually harming your daughter in some way, she doesn't get to unilaterally control access to her, or use her as a pawn or a bargaining chip to score points against your mother. She's your child too. You have some agency here.
Stop trying to mediate between them and stop trying to put over your mum's POV to your wife. That's a pointless exercise. Even if you truly believe that your mum is the innocent party in some of these disagreements, your wife is not going to change her opinion of your mum. She will just accuse you of being disloyal and under your mother's thumb. Accept your wife's position and respect it. But take control of this now, so you don't spend your entire marriage in conflict over this. Tell your wife you will never again expect her to spend time with your mum, or communicate with her in any way.
However she also needs to respect your position. You do still want a relationship with your mum, and you want your child to have one too. She can control the relationship she has (or doesn't have) with your mum and that's fine. But she doesn't get to control the relationship you have with her, or the one your child has, via the connection to you. That relationship can function perfectly well without any need for any involvement from your wife, so long as the three adults here all agree to behave like adults. Take your child to see your mum as often as you like, within reason. If your mum is farly local then take your daughter for regular, short visits. If she is quite a distance away then facetime your mum with your child, so you can manage longer gaps between in-person visits. Don't allow your wife to dictate to you how often or when this happens. It's really none of her business.
If your wife refuses to accept this, (unless there is some very damning information about your mum that will be drip-fed later) then the person being selfish, controlling and manipulative here is your wife.
With regard to Christmas Day, even if your wife agreed to host your mum under sufferance, there is going to be a terrible atmosphere all day and surely you don't want that? It's just not worth it. Equally, it's not nice knowing your mum will be alone, so you need to find a compromise and your wife needs to accept that compromise.
If you want to see your mum at Christmas or very close to it, then see her. Your wife does not need to come with you, but you certainly do not need her permission to go, or to take your child. If your mum is reasonably local then you could visit for a couple of hours on Christmas morning while your wife gets the lunch/dinner ready. Failing that, visit her on Christmas Eve or Boxing Day for a good two or three hours. Have a meal with her, take some of your daughter's toys with her so she'll be content to stay a while. Don't dash in and out. That still leaves plenty of time for your wife to have the Christmas she wants, which excludes your mother (but no doubt includes hers.)
And your wife should accept that the price she pays for successfully removing your mum from her life, is that you will sometimes need to split your time at weekends, Christmases, birthdays etc, to take your child to visit or spend time with your mum at times that are important too.