Some people here seem to be of the mindset that by continuing a relationship with them is a form of pursuit.
It is tricky having a close relationship with the parents of your W, when the M is on the rocks. First of all, they are her parents, not yours. Even if she's in the wrong, they eventually will support her b/c she is their child. You may be an adored son-in-law, but you are not their flesh & blood. Secondly, as you have seen, the W is extremely jealous of her parents, and I'm sure she feels that you tried to steal them away. That's why she decided to take whatever means necessary to confidence her mother to side with her. Thirdly, subconsciously, I think the LBH is wanting her parents to influence her back into the MR. That's understandable, however, it doesn't work. And you give various reasons why you told her mother or why you want to remain close to them, but at the very bottom of your heart....it's her. They are the next thing to her. You are desperately wanting to make some kind of a connection with her.
We advise the LBH not to discuss the a lot of details about MR....or seek out advice from his wife's family b/c in the long run, it can cause problems with them and you either during the reconciliation or afterwards. Maybe not in every case, but why take that chance, especially when you've already seen how badly it can turn? I would say to remain warm and friendly toward them, but let them initiate contacts, especially her mother. I don't think you should be dropping by to visit them while things are not good between you and wife. If nothing else, it probably would cause some awkwardness at this time.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!