Your work trip to France being sabotaged by his ineptitude is infuriating. What has your work been like about it? Have you been able to make alternative plans?
This man is still bringing stress and disappointment to your life.
I understand that you want to keep him on side and your decisions are made with your children in mind. Of course it makes sense to want them to have a good relationship with their father.
He's the one who's let you all down. His chaos and unreliability is the opposite of what you, the children, the grandparents need at the moment.
I wouldn't try to support him with his housing as he's let you down. He doesn't deserve your help and he's on £50k so he can sort himself out. I get that you're thinking of the children while they're with him but honestly, his attitude doesn't suggest that he'd be respectful of your generosity and put the children first.
With his getting the log book, I wonder if it's something to do with a police thing involving his car? A driving conviction? Or maybe he's planning to go to France where he needs the info from the log book for that thing you need to do before driving over there (can't remember the details, we had to do it for a trip there this year).
But yes, it needs to be a clean break with none of his stuff in your family home. He's too comfortable with the current set up and it's not going to work for you.
I get what you mean about maybe the admin and legality of divorce being a bit much but I've got a male friend whose ex really dragged their divorce out. They'd been living separately for years, he met someone new, and they couldn't buy a home together until the divorce was completed. She was in their old big family home, he was in a small rental. Anyway, the delay can't drag on for too long. If he's already got a new girlfriend, he's not got a leg to stand on in terms of delaying in surely.