I rarely have contact with my ex but our daughter got married recently so we saw each other at the wedding. He said, where lots of people could hear, "I paid for the wedding." Omg is that so absolutely false! It's laughable. Yes he contributed but far less than our daughter and her husband and far, far less than me.
I would have loved to set the (financial) record straight. But I didn't. The truth is the truth and if he wants to make himself look better by lying that is his problem and thankfully, his problems are not mine any more. The truth always comes out in the end, one way or another.
Your ex is trying to make himself look better by lying. So maybe this new woman, or others who see his FB posts might believe he's this great dad when the opposite is true. The people who matter most - number one being your child - know the truth. His mother knows it, too.
I think your handling of his mother is great, btw. No, she can't be left alone with your daughter but obviously she cares enough to spend time with her and you make it possible.