She's very young and has a baby. She probably is leaning on her mother quite a bit. It's her mom. I know this may not be what you want to hear, but it's possible your son doesn't mind as much as you think. You sound pretty aggitated by the situation and even if you have not been saying anything to him directly, he is your son and he has picked up on it.
He may have told you that he said something just to make you happy. My husband did this with his mother for a while. She was not happy with the way we did this or that and when she would talk to him he would say 'I know I tried to tell cpr, but that's how she wants it' just so his mother would leave him alone. It ended up backfiring when a conversation turned to something that she and I actually agreed on and he was using me as an excuse! Now he sees how that created a lot of tension between her and I and he wishes he could take it back. Your son is young too. He probably still wants to make you happy and proud and may feel just a bit guilty not taking your side as you have been very generous.
As far as her mother being there for every visit, have you ever thought that you may make her a little nervous? I noticed that when you spoke about their home you called it your son's home, not their home. You may not even be aware of how a little thing like that could intimidate a very young mother. I know that you purchased it, but she lives there with your son and they pay rent. Even if he is working to pay the rent up until now, she has been taking care of your GC, birthing your GC and growing your GC within her body. If she feels like a visitor in her own home when you come around she may be asking her mother over for moral support. It sounds as though you may not be hiding your feelings as well as you think.
I may be wrong, but I do hope that you will give these possible scenarios some thought. This is not just the partner of your son, this is the mother of your grandchild. Developing a respectful relationship with her is in the best interest of you, her and your GC. This relationship with your son may or may not last. I hope that it does, but should it not you do not want to be part of the reason it didn't and you will wish that you had a good relationship with her. My cousin and her husband had a similar situation and it did not work out for their son, but they always stayed on good terms with the mother of their grandchild and respected her role in raising that child. They now see that baby more than their son (unfortunately he does the bare minimum). Because of the respect they showed to her and with which they treated her, they feel very comfortable calling her when it is not their son's weekend if they have something special they want to do with GC. And so far she has never said no. For everyone (including yourself) I would say you should 'butt out'. If not for your son, if not for you then for your grandchild.