I think the dad needs to invite his daughter on the trip. He needs to include her, demonstrate that he loves and values her.
She is being invited. There's that covered.
Her dad, not the OP, should pay for her ticket, bookings etc.
So he should pay for one of his children, when he's not paying for his others. The non resident should be treated better than the resident children. Zero for DC1, zero for DC2, several thousand on DC3. The trip is only happening because OP has received an unexpected inheritance. The father's part is the same to all his children.
Include his daughter in the plans and discussions for this wonderful trip. Plan how to accommodate everyone's likes and dislikes (rides etc) as fairly as possible.
This will all be part of it if the invite is accepted.
If the worst comes to the worst and daughter backs out at the last minute then dad will unfortunately have to take a financial hit.
No. The mother can. Because if this was a resident child, they wouldn't get that option. But the mother can allow this at her whim.
It's terrible, I know, but his relationship with his daughter won't take a hit and that's far more important imo.
If the daughter doesn't go, then it's got nothing to do with the father. So their relationship won't take a hit. The invite is there. The mother just has to provide assurance that she's not going to be a dick this time, because many people will be upset and thousands will be lost.
I don't think OP's plan will work. Hoping I'm wrong, but she doesn't think it will work either (95% it won't she said) so that's not good.
But after ten years of this woman doing this sort of behaviour, this is all they're left with. Both father and child have to suffer the consequences of the mother's behaviour. The person with actual logistical control of DD is unreasonable. This mechanism wouldn't be in place if she was. OP could just book everyone's ticket in peace.
I don't think this plan will work either. Because if the mother pulls a stunt, for the first time, it impacts her. It tells OP that the stunt was indeed coming though, so she's just saved herself thousands, and not having to upset her children when they discover on the day that their sibling won't be coming anymore.