It's a three week luxury holiday that it takes two working adults three years to save up for, so presumably we're talking thousands, potentially five figures per head...doubtful that the boyfriend/daughter would be able to fund it by simply 'picking up some overtime', regardless of notice, unless they're high earners.
I'd just be honest with your daughter OP: You'd love for her partner to come and he'd be very welcome, but you're not in a position to help fund it. Tell her that you completely understand if she doesn't want to go without him, and let her know when you intend to book so she can have a think, talk to him and give you an answer.
If they're moving in together soon it'll be slightly more awkward her going away for three weeks without him. Could your daughter go for a week or ten days?
The 'struggled being left' thing is a bit of a red flag though. In what way? Assuming this was three years ago so he'd have been a teenager - was it just a bit of love-sickness? Hopefully he's grown up a bit now and this won't be an issue?
But if he's making your daughter feel guilty about spending time away from him, or 'choosing' her family over her partner, that's something to keep an eye on.