J.
B., part of the answer to your problem has to come from your husband for it he that must take a stand as a man and defend his family even if it's against his mom, dad and siblings. His first family is you, the child and he and everyone else is second. He needs to hear both sides, your's and your sister-in-law and ask his sister why she is being so critical? I had a similar situation with a relative of my wife and since this relative was very close to my wife as they were growing up, I address the situation with my wife and informed her that I was going to confront this person.
My wife chose to address the situation first before I could say anything, the relative did not take my wife very serious and still chose to make unkind comments about our children.
We were invited over to this relatives house for a family get together and she chose that moment to take a jab at me to which I responded honestly and expressed my opinion as to why what they were saying was so rude and stupid.
We were not invited back to that relatives house for nearly 5 years which was fine with me and my wife. That relative doesn't continue making rude comments about my children for she knows were we stand, she has never apologized but now she wants us to spend more time with her family to which we usually say that we do not have time.
This relative of my wife has the same problem your sister-in-law does "jealousy", her children do not get all the attention when we go to family reunions, mine have been great students are friendly and polite while hers have been rude and brats.
Our children are grown now and off to college and one ( our eldest son)has children of his own and this same relative of my wife that I've been talking about, started having children again so her youngest is the same age as my granddaughter and just like before she started making her rude remarks but now it was about my granddaughter.
My eldest son quickly solved the problem by putting her in her place and just as I did it was not done in a rude way for he simply informed her that it was not her business to get involved in his life and if she had something to say to him or about his family she needed to approach him and say it face to face and if she can not then she neeeds to leave thenm alone and mind her own business. This happened two years ago and since then this relative has been trying to be as nice as she can be when she is around our whole family. WOW! This is a very long story but the point is this person could only be set in her place by a firm man that took a stand to defend his family and since her husband does not have a back bone and caves in to everything she say's she is not used to a real man defending his family but once she encounters one she backs down. I don't mean to offend anyone and please do not take it that way, but this is a problem that your husband needs to atke serious and defend his family honor.