I wouldn't be canceling trips and days out for bad behavior because you'll just get into a cycle of: they do nothing > they get bored (I don't know why there is a line through this) > they behave badly --> they lose their next day out. Lather, rinse, repeat. They need a different consequence. (Notice how canceling their day out means he doesn't have to take them all out for the day and deal with their behavior when they are out?)
Also if they are overweight then canceling their days out means they are going to get less exercise which will not help their weight/health problems.
Also, I think the nastiness to each other won't extinguish unless he gives them different ways of dealing with conflict. So maybe a punishment would be they have to find a way of saying the same thing with less inflammatory words. Then he can go over their words with them. For example, if Kid A calls Kid B nasty names, then he sends Kid A to their room and then after 5 min he goes into Kid B's room and talks about, what happened here. Then go into Kid A's room. Ask same thing. Then, tell Kid A, here is how you approach this situation. This is what you say when this happens. Kids don't automatically know how to deal with conflict. They have to be taught. Then the consequence is, you have to go apologize to Kid B.
If mom neeeeeeeeds the 4 days with the kids then this is a bargaining chip for him. Because his other option is to ask for full custody. (Then they would be in a consistent environment and he could really work on their behavior). So he could say, for the next few months I would like to only have Tim (whoever is the oldest) for the weekend. Then Sandra only for next weekend. Then Bella only for the weekend after that. And then when that one kid is there, he does father-child things with that child. Alone time with their parent is what children generally crave - your total attention to listen to their Pokemon ramblings and why Mrs. Smith at school is mean. Alternately, he can take ONE child out each weekend for a hot cocoa, giving each child a turn one weekend out of the month. My kids' behavior was so changed by just having time with their dad's full attention. If you can't handle the other two kids while he takes one kid somewhere, then that is why he needs the children's mother to have them for the weekend so he can focus on one child at a time. Again, he has more control of this than he thinks.
With my kids, they do tend to argue if they are all cooped up together for too long. So I spend a lot of the weekends driving this kid to this friend's house, that kid to that friend's house, this other kid has a friend over, and so on. Just keeping things very active and a lot going on, and again, if two kids were out at a friend's house, the other kid has alone-time with their parent. "The Devil finds work for idle hands," which basically means if kids have nothing to do, they will FIND something to do, and it will usually be to annoy you, or annoy each other, or break something.
Sorry, I didn't speak to the issue of you moving out but those are my thoughts on how he should start making progress with his kids.