Looking at this from a logistical point of view, removing all sentiment etc, it's just not FAIR on the GF to have a 2yo to care for out of nowhere. He's a CRAP dad and it will scream that to his GF loud and clear, I'd be surprised if she didn't dump him before New Year.
I actually agree with some of what AThing says
"The OP wants her daughter to have a relationship with her father.
His distance from his daughter is a result of his choices and preferences.
My advice is that until he is happy to change his own toddler's nappy he shouldn't have her overnight, and apparently that is crazy lunacy because obviously it is great for a 2 year old to spend a weekend with a total stranger changing her nappy."
I think OP, that you would be best served by coolly telling that in principle of course he needs to take an equal role in looking after his DD, but that due to numerous different reasons, he is not quite ready to take over that total responsibility and that it's simply not fair on ANYONE, least of all his DD that it will fall to his new GF. It's not a jealousy/bitter thing, it's a sensible parenting decision thing.
Remind him of the park poo drive by nappy, not to have a go, but if he can't care for her yards from your home, what makes him think he can 300 miles away.
Tell him that you would be the first to encourage that he takes her and spends a great christmas with her, but that atm there is too much that he will struggle with. Tell him that this time next year, she will probably be pottytrained, so there won't be the issue that there is this year.
Suggest that THIS christmas is YOURS and that in the next 12m you will BOTH work towards HIM having her overnight, and ideally for christmas.
There is no rush, your DD is 2, next year she'll be 3 and it will be easier.
Say NO to the christmas visit this year, and state that he can have next year. If he feels strongly enough, he can take you to court... it won't make the court list before christmas I shouldn't think.... and if you are sensible, calm and pragmatic, no judge would ever rule in his favour when next year is offered instead.