It sounds a bit different than what I'm going through in that we have no kids involved and the only real weight issues with my wife are in her mind.
However, it's similar in the fear of aging and need for assurance that they are desirable. In my case, it's so odd because so many people constantly compliment her that I wouldn't believe she could have issues. But such is her outright fear of aging. She openly says she'd rather be dead than get old, not something you want to hear when you always imagined getting old together.
But there's also a thing with her where it doesn't matter what I say about her looks, I'm her husband and she thinks I view her differently and would find her attractive no matter what, so it doesn't count.
I think you're going through that with your wife and I have no idea how to solve that problem.
All I can say is it seems the best way to work with their feelings is to find ways to reconnect as you and her rather than dad and mother. I know the "dates" we've gone on upon the recommendation of a counselor have done some reminding of why we're together in the first place. If it's possible to do that (I know you have kids), it could help satisfy some of her need to get out while being supportive of the two of you as a couple.
It's likely that she needs to get away from you and the children because she feels burdened down, but, if she's willing to compromise it could fill both of your needs.