Here's the problem about people like this: they make others think there's a problem.
It's very hard to prove a negative, and why should you have to prove it just because these people are determined to undermine you for their own weirdy power games.
Why is it acceptable that they can question your sanity to the point where you somehow have to 'prove' you're not ill? Or force you to 'prove' your fitness to be a parent, which sounds like its under question for no particular reason?
The way they are behaving is disgusting. And it's certainly not the way to behave if they really had grounds to think you needed help. Screaming battles where they use any perceived issues as traction to help themselves in a position of control and dominance over you? Not exactly conducive to good mental health. But would allow them to push and push until you are in a heightened state of anxiety and panic.
Not benign or just worried, from what you've said. Gas lighting and very unpleasant indeed.
Your reaction to them being horrible and not in the slightest bit helpful is to defend your boundaries. That's the right thing to do, even if it feels very hard and they push harder and harder. Having no key keeps them at bay. Giving in would allow them to continue their campaign of awfulness.
On a completely separate topic, if you find your anxiety doesn't go away once you're not fighting them for your basic right to have privacy and self agency in your own home, then maybe go see your gp?
And the same about eating. I'd be very careful about what your own relationship with food is teaching your children. But then, I'm not sure how difficult you find food as you mainly comment on your in laws reaction which is hardly the most balanced perspective! But I would keep an eye on it, but takethis topic far away from the whole nightmare in law thing. It would be easy to get so focused on their awfulness that you end up letting things drift when there is help available. I'm not saying this because of a 'no smoke without fire' effect of your in laws terrible behavior, but as a totally separate thing in case it's something you want to think about after dealing with the in law situation.
But dealing with them is a priority. You need to protect yourself against them, and if that means not seeing them then yes, I think it's a good idea. Though I'd question whether that will really help if they're that determined to be spiteful and cruel. Will they start to demand your dh & children celebrate special days with them and you get excluded? Or will they carry on dropping poison into your dh's ears that he can't deal with? Especially in front of your children?
I think you and your dh need to think carefully before you do anything. And he needs to focus on protecting you from situations like this one. And whether he's giving the wrong message by letting them push you out of the family by excluding you from seeing them?
Good luck