You don't like your MIL, you feel she is possessive and you dislike that she expects to continue a close relationship with her son, as she has with her daughter. You don't like her and so you want to limit contact with her. This limitation then extends to your first child, her first grandchild.
Limiting her access to your child won't make your child like or love her less. Generally speaking, children's lives are enriched by having adults around them who genuinely love them. This may be hard on you, especially if you have intimacy barriers that are triggered by close relationships, or human interactions that are not on your terms. From what you've explained, your expectations of a loving family are different from MIL/DH/SIL. You have owned that. This doesn't mean they are wrong or that you are. It does mean that your feelings need to be owned by you. It is you who are finding closeness with wider family intolerable. It isn't that any of their behaviours would be considered unhealthy, unsafe or dangerous. You just don't like them for your own reasons. Is this enough to limit your child's relationship with his gran? S/he may resent you in the future, if you do limit them. Grandparents don't last as long as you might think. Your DH may also resent you, it's his DM after all..
You are part of two wider families and you are choosing to distance yourself from one of them. Your child will still be part of their clan. Your child will see that and feel that. They will also see and feel that you have a problem with this. This is likely to confuse them at best. How will they understand the message that you will be generating?
We all make accommodations and compromises with people in our lives. You have boundaries and that's fine. But be clear, it is very very natural for grandmothers to want to play a part in their gcs lives. If this is a problem it needs to be managed by you, because it is your problem. Tolerance, empathy and understanding are probably something you will expect from your child. It needs to be modelled first.