Ah bless you, you are sadly in a position where you have to choose. This is no-one elses decision as its your house and your family.
So lets look at your family. I haven't seen how your mum came to live with you, but that does have a slight bearing on it. However a man entered your very well established life, being you, mum and kids and presumably it worked well, on the whole. You start seeing a guy, he meets said mum and kids and knows what he is getting into but in fairness the reality is sometimes different to expectation. When living apart you do put on a front, your best behaviour and that cloak usually disappears when you live together. I may be wrong but its your home, he pays his way but does nothing else? When he pays his way does he just pay 1/6 of the expenses, cos if he wanted to seriously be part of this family as opposed to a lodger, he should be paying at least a third if not half of the expenses, as surely if he wants to be the man of the house, he needs to take a modern mans responsibilities? That includes not just paying his way, but taking his share of financial responsibility for the family and maintenance, cleaning of the home and childcare. Your mum is part of that family by birth, he can't make demands of this nature as, lets face it, he is a lodger with benefits, not really a family member as a family member wouldn't oppress the family in this way. However, be honest, how easy is your mum to live with, is there room to accommodate both by some adjustment by each of them? If the answer is your mum is good, he had no right to make this demand, no right to be joint decision maker as he does not accept half responsibility. He wants his cake and to eat it.
Personally speaking, I don't think he has any decision to make, its not his choice, its yours. He has given you an ultimatum and if you do not throw your mum out, this relationship is downhill from now on in. That's reality. You have no option as he has no right to make this ultimatum so he needs to go, not your mum.
At some point in your life a man may come along and he may take a guiding and fair share in your parenting and financial role and he may love mum but at some point (sorry don't know her age) she may need to leave, but thats a joint decision not bullying tactics which is your current partners mode - big difference!
If there is no room for a liveable compromise, YOU, need to tell him to go, as you can't inflict the ensuing paddy on your family when he realises mum is going nowhere.
To be honest, the fact that he has put you in this situation, speaks volumes about him, and its not a good look.
You knew what the answer was, you just needed validation, you have pages of it here, so go do what you have to! Its going to be tough but you know what needs to be done!