AIBU to object to ex hosting family barbecue in my home?

No, YANBU. He doesn't get to stay in your home. Allowing him to do so has blurred the boundaries, and this is the result. Dial it back OP.

He has now said that while I’m away he plans to have his family stay for a few days and host a barbecue. Knowing him, I worry it could turn into a bigger gathering with more relatives and possibly neighbours.

No. A big fat NO. This is now your home, it's not his any longer. You're paying the mortgage, not him.

What on earth does he think gives him the right to announce (not ask) that he's going to have his family to stay? Not even just to visit for a barbeque but to stay???

This isn't about the DCs, he's just tried to dress it up that way. What he wants is somewhere to entertain and host.

When I raised concerns about similar issues in the past, he’d say it was my issue and make me feel unreasonable. So I’m wondering — am I being unreasonable to feel uncomfortable with him hosting his family at my house while I’m away?

So he turns it on you. You are not unreasonable, probably never were. And if you'd become distant and unloving I can hazard a good guess as to why. His constant attacks would kill any sort of love or affection or closeness.

Tell him the arrangement is off. The DCs will go to his. And if you haven't already got one, make sure you have a Ring or similar doorbell before you go away. If he is joint owner of the house I don't think you can change the locks.

If the house is 100.% signed over to you and you’ve bought him out and are paying for it all then yes YANBU

if he owns half of it STILL then he will see it as half his Perhaps???

Well, I don't think you're being unreasonable to feel that way, but having allowed him to stay in your home (his previous home) while you're away, seems like a bigger deal to me than him having his family around for a BBQ. So to me it seems a bit petty to object to that if you're going to let him stay.

I used to have people house sitting quite a lot (preferred it for my cats to a cattery) usually young people I knew who enjoyed having the house to themselves & loved the cats. I always said they were welcome to have friends around for a BBQ or party as long as they were mindful of the cats & neighbours.

Different to your ex I know, but really, if your letting him stay, dies it really matter if he has his family visit when he has the space to do so?

How horrific.
This is totally unacceptable. Let him know that he won't be staying at yours.
Sorry, but this is a huge breach of your privacy and i imagine, may create a difficult situation for children too.

No, his family cannot stay in your home!

I would find that unacceptable. There’s a huge difference between him being there, and a large group of his guests. If there was a problem- ruined carpet, broken lamp, bbq fire, it would be really hard to get past.

He can take the boys to visit his various relatives across the three weeks, and leave your house out of it.

Is the house legally your now and the finances settled ?

I think it would be fine at this point to tell him the deal is off. The boys need to take what they need to dad’s , then you lock up and don’t give him a key.

This is beyond CF territory.