Pink, that does sound awful and I can relate to a lot of your post as a Mum to my own child and the way his father has behaved.
However, there is so much that your ex could do to make your DC feel loved and valued by him that doesn't involve overnight stays. I can see that being pushed out of the old family home must be very hard on the kids but never taking them out for the day on his own, having them round for tea, insisting on taking step siblings out etc. is really what is the problem, basically he's being an arse! Overnights or no overnights.
My own DS stays overnight at his Dad's (no bed or belongings are allowed to be kept there), does it make for a better relationship with his Dad? I really don't think so. I think it just reminds DS that he's not really part of his Dads life all the more. It also makes it easier for his Dad to say he has a "relationship" with DS when he can sit in another room and ignore DS for the weekend, whereas if he had to take him out, he would be forced to talk to him and interract with him.
I think it all depends on so many things, closeness to PWC home, school, friends, how many hours NRP works etc etc.
I have seen my 4 DSC when they were younger literally being dragged out for hours in the car to faciliate 4 different school runs which are miles away (had they lived with us, they would have got free transport) rather than getting 1 lift home with Mum / getting free transport back to Mums and Dad picking them up from there later. I've seen my DSC being made to get up 2 hours earlier than they would at Mums to fit round DP work and the school runs. I've seen sick kids being transported around by car to faciliate the rota and timetable. I've seen kids getting upset and stressed by having to remember their Monday morning books and clothes, show and tell items on Friday mornings knowing that they are "not allowed" at the other parents house until after school on Monday.
Who really benefits in all those circumstances? Dad gets to say he has his "contact", Mum gets a break, the kids get stressed and bored when it's not necessary. I know that every situation is different but I actually believe my DSC would be better off only staying overnight at ours on Fridays and Saturdays and school holidays.
And as an adult who was passed from pillar to post as a child, I really feel that kids need stability and routine. All I wanted was a permanent, safe place where I could feel at home and I could be part of another place but I had the security of my home when I needed it, not when a court/rota/parents agenda dictated it.