Call his bluff.
Do you have any joint friends who are male (male is important as he's less likely to kick off with another man) or a good friend's husband? Say his name is John, have a little chat with him and then set up a WhatsApp group with your ExH, John and yourself.
Call the WhatsApp group "Monday Nights" and send a message saying "ExH, John has kindly offered to look after the DCs on Monday nights when I start my new job in Sept, so he'll be picking them up from yours at the usual time, bringing them home and staying with them. I figured it'd be easiest if we all were on the same group chat to communicate about the details, or if anyone is running late etc."
Your ExH is likely to message you directly, kicking off. In which case you screen shot his message, put it in the WhatsApp group and say "all correspondence about Monday nights from Sept onwards will be going into this group chat."
Then answer whatever ExH has said in the group chat "yes, I know you don't like the idea of John having the kids overnight but you said you wouldn't have them and I won't be home on Monday nights come Sept, so you didn't leave me much choice."
John can also add in "I agree mate, seems strange you'd prefer another man to look after your kids overnight than have them yourself, but OP needs to pay the bills, so it's either me or you looking after the kids."
I would bet he backs down, but worst case scenario, John has to pick up the kids a few times and drop them to yours where the sitter is. I doubt he'll let that go on for long - it's embarrassing for your ExH to hand over his kids to another man to care for simply because he can't be arsed.