It would appear that she is as sick as her abf is. If she continues to stay with him, that is her decision, at the same time that does not mean you have to interface with him or listen to her crying and complaining about him. It is apparent that she is all drama and no action. If she wanted to leave him she would, she doesn't. Her actions speak for themselves.
You do not "have to be" grateful or thankful to him, your mother is attempting to manipulate you, to get what she wants. To play both ends against the middle.
As a parent, your first responsibility is to your child, not her, not him. Personally I would not expose your child to their toxic relationship. If you do not want him near your child (which is a responsible decision) and she insists that you must accept him with her...then...you may have to make a decision about your relationship with her. If her priority is him, over you and your child, then so be it....it is her choice...it is your choice to continue the relationship with her or not.
As a child, I lived with a mother (an alcoholic) who ALWAYS put her abusive men first, she never had her priorities straight, and, her children suffered. I am no contact with her once again, the third time in my adult life.
Do what is best for your child, you are your childs future. Children carry their childhood into adulthood, they are the true victims. Just as you are a product of your childhood, your experiences, so too, will be your child.
Keep posting, we are here for you.