Why does the OP need to "make a stand"? It is just the reality of having separated parents (or even just being someone with a lot of friends and an active social life) that events will clash occasionally and you have to choose which one you will attend and then politely decline the other invite rather than try to attend both by asking the "host" to change their plans to accommodate you.
Her Ex is planning a family ski holiday in the only school holiday that is suitable (Feb half term) and has invited DD along. It's not his fault it clashes with the OPs birthday and he has no obligation to change his arrangements so DD doesn't have to choose which event to attend.
He is also taking his other children, and presumably his new partner. Why should everyone's holiday have to be cut short/dates changed/extra plans (eg picking DD up from the airport at a later arrival date, which will take more logistics as ski resorts tend to be at least an hour or two drive from the closest airport) to accommodate her wanting to do everything rather than her accepting that she can't be in two places at once?
I share custody of DD with my ExDH. We have been split up since she was tiny so it's pretty much all she remembers - there have been many times where events have clashed, sometimes we have agreed between the adults which event DD will go to due to logistics/priorities. For example if ExDH wants to go away for a random weekend but it's the same time as my sister's wedding, the wedding would take priority; if ExDH has been offered extra tickets for a concert by a friend but it's the same weekend I was planning to take her to a theme park the concert would take priority as tickets are already purchased etc and it is presented to DD as "Dad has tickets for you to go to X next weekend" with no mention at all of the other option. If we had discussed it already as a possible activity and DD said "What about Y?" Then I'd just say "Oh we can do that another weekend, unless you don't want to go to X"
Other times it's been a case we have told DD "Dad is doing X, Mum is doing Y, which do you want to do?" and then whichever option she has chosen we make happen, but we have never twisted ourselves in knots to make sure she never misses anything ever not have we ever made her feel guilty for choosing one parent over the other.
In this case I would have told DD "You can either go skiing at half term week with your dad and we'll do something fun for my birthday before/after the trip or you can stay here for my birthday and we'll go away for a few days somewhere in the UK. Which do you want to do?" The possibility of doing both by changing the times/dates would never have even entered her head as it isn't an option on offer.