Leaving his depression aside for a second, I think he's perfectly reasonable to ask to see more of his kids. I assume he loves them just as much as you do. Currently he goes a week without seeing them ( Thu - Thu ). It is in the children's interests to have regular, quality contact with their father, and their gran.
It would depress me tbh if my XP said she wanted to stop the midweek sleep over. And I think you do need to keep in mind that for him ( and by the sound of it his family ) your suggestion of less time is only going to fuel their resolve to push for more time, they'll see you as the bad guy, and it'll damage any amicability that exists currently.
The issue of dc's behaviour when they return is I think a very tricky one. My XP ( as I'm sure a lot do ) also comments on this and it's difficult to know what the solution is as we're two different people, who couldn't make the relationship work, who have different styles of parenting.
Add to that the fact that by virtue of seeing my dd infrequently the time with her is 'special' and has a very different rhythm to the weekly humdrum and stricter schedules that she experiences with her mum.
There should of course be discipline when with dad, it can't all be ice cream and fun in the park. I'm just trying to say that to a degree the behaviour is / can be a product of the situation of being a split family, not necessarily all due to dad's lax attitude to dc.
Bringing his depression back into the equation... well, that does change things a bit. Especially when you say he's not allowed to be alone with the dc. I don't want to pry but I assume something worrying happened in the past to cause this situation.
At the end of the day he is their father, he loves them too, but it's in everyone's interests , especially the dc, that you are both able to work together now and in the future. If he has mental health issues then you are perfectly entitled, indeed have a responsibility to keep your kids safe whilst encouraging a decent relationship with their dad..
It does sound like you are trying to do the best thing. In the long run I think you should be thinking about him having the dc more and it may keep things on an even keel if you suggest you're open to the idea but include some caveats. He needs to deal with his depression, perhaps his parenting style needs to be improved, it needs to be worked up to gradually......