Tricky situation

I know you say you're all only in the city for a few hours - any wiggle room on that? Have a wetherspoons type lunch with him (they have an app, so he can order and pay for himself like a big boy) and pretend that you're all finishing at 3pm but then go for an afternoon tea somewhere fancy just the three of you? Intersperse it with a trip to a museum or film if you like. I don't even think that needs to be kept a secret, it'll just be "Oh this has been lovely hasn't it, now I've booked a table for me and the girls for afternoon tea for later on so we need to go, cheerio!"

I think you're possibly correct quite correct about confrontation being unavoidable eg OP advises ex in advance of travel to event that she won't pay, he might turn around and say to DC oh, i can't afford it now so can't come. OP advises on night, some sort of scene or atmosphere spoils the evening.

Apologises if this has already been suggested, but when the bill comes, divide the figure between number of people present then work out his share. So if there's six of you, ask him for a sixth, then you cover the rest. Tell the waiter the exact amount you're paying when you hand over your card and then point him towards your ex to settle the rest. Hopefully he won't argue with that.

The problem is it sounds like you are making the meal all about the beef you have with your Ex H (which may well be legitimate) and not about the actual celebration itself.

So you run the risk of turning the celebration into a mess because your Ex. H is an arsehole. How that will pan out, whether your kids will blame you or your ex. H for it is hard to determine.

I'd ask myself the question whether it's really worth it.

You can come up with all sorts of ways to separate out the costs and the restuarant will almost certainly accommodate you. For example you could ask the restaurant to do a set menu for a set number at a set price and pay in advance. But probably anything you do is likely to lead to some confrontation if your Ex. H is as tight fisted as you say he is, and that confrontation is likely to bring the celebration down.

I would probably pay, and have a quiet word with your kids that in the future if they want to celebrate with their father separately then that's fine but not together with you.

You must have been reading my mind, I was just about to say exactly the same thing. Let him do either lunch or dinner, and OP vice versa. As you say it is a different story for life events.

The adult child should have had more cop on than to put this unnecessary stuff on OP. Bet he is not one bit worried about anything.

When you see him at the meal, ask if he wants to split the bill 50/50 or whether he just wants to pay for what he has. He is the one who will come across as petty if he doesn’t split it.

OP avoid any suggestion of 50/50 as some posters have suggested because then you will end up sharing the extra expense of his booze.
How did your DC word their invitation? It may not have sounded like you were hosting and paying for the meal. If ex is disabused he may decide not to bother to attend.
I this event is to celebrate a graduation then how galling for you who has assisted with fees and support to have him muscle in on reflected glory.
Good luck with how you manage this.

How can she text him they dont have any contact

This is even better though

Yes I think this is good, and I’d say it before you for the meal if you have his contact details.

If you wait til the day you might get “oh I was told this was your treat”

”Hi X, I’ve heard you’re coming along to the meal on the Y date. Do you want to go 50:50 to treat the kids, or will you just be paying for yourself?”

Ideally by text so there’s no room to deny it had been said.

Then on the day, ask the waiter to split the bills as others have said.

Thats the thing if he has form the special event is going to be ruined by him so wont even be worth it

Based on what you’ve posted, I think you are naive to think that he will come and pay for his share.

I wouldn’t put myself in that situation. You want a nice meal with your kids-it Doesn’t sound like that will happen if he comes. I think your child should arrange to see him separately.

Just tell him. If his feelings don’t matter then what difference will it make?

That’s a conversation to have with your child then.

If I were you I would leave my money at home and let him cover the bill.

I presume it's a graduation? Nice for your DC to have their dad at that, but I can see why it could be really awkward for you.

I'd ask for the bill at the end, ensure it is given to you and then mentally take off what he has eaten/drunk .... tell the waiter you're paying x amount and ex DH is paying the remainder.

I'd literally just put a smile on my face for the children and just pay, if he offers fine if not fine too. Its a one off & life is too short. I would avoid making a landmark birthday being about who paid for what. Your children will thank you for it & will already know the bigger picture. I think the fact the child invited him speaks volumes & shows they are hoping things can be put to one side and the event can be about them & their celebration not who's paying for what.

The OP said the kids are adults and already know everything.

There isn’t anything a nasty about telling an adult, who invited someone else without asking the OP, that the extra person needs to pay for themselves and since the adult child invited them, they can be the one to pass that message on.