First, can your mum care for your child in her home and not yours for a while. Not for your cockwomble husband's sake, but for her sake, as walking into his face as she does her unpaid and unappreciated childcare three days a week must be pretty upsetting.
Then I'd have a chat with him and lay it out like this: 'yes, my mum is involved a lot in our lives, but that's because we chose to have her care for our child, for free, three days a week saving us several thousand a year in nursery fees, plus having her do babysitting, date nights and so on. If you don't want that any more- let's put our child into nursery/pay a babysitter, and we can have our house to ourselves. If you want her to carry on, then you have to shut up and put up, because you are essentially abusing her good nature in doing this free work and it's not fair. Either treat her nicely, or let's stop the work she's doing for us and have a more distant grandparent relationship where we just visit and she doesn't care in our house'.
I'd also add 'I love my mum, and I know she's a nice person, so every time you slag her off, you are hurting me, if you carry on, it will seriously affect how I feel about you. It already is'.
He might well feel overwhelmed in his own home, in which case, this will give him a chance to get out of this situation by paying for additional childcare and resetting the relationship.
No-one is nasty about my mum though, she's amazing, does childcare, bought the children so much, cared for them in emergencies, helped my husband with his career, and whilst I don't expect him to think she's a saint, and don't mind the odd moan or him
bodyswerving the odd family event, I won't take full scale nastiness either behind closed door or to anyone's face when they have been helping our family so much for the past decade- and nor should you (and if he started treating my mum like a piece of shit on his shoe, I'd leave).