Honestly? I would pay it.
Regardless of the ins and outs of whose diaries and commitments needed greater consideration and prority in the planning of the trip, and who has a right to feel slighted etc, no-one held a gun to your head.
You agreed to go on the trip, now you can't go on the trip. The reason you are no longer going - the situation with your mother and the childcare - is not their fault or problem, so the onus should, I think, be on you rather than them to cover the costs of your cancelling.
I know you were cross about their lack of consideration when organising the trip, but that's actually a seperate issue I think.
Put it this way - if you had helped plan the trip and had been happy with all of the arrangements and had been really looking forward it but then you had to suddenly pull out you'd have offered to still pay your share without question, wouldn't you?
You can't punish them for not being as thoughtful and considerate of your circumstances as you'd have wished during the planning stage by now not paying your share because you have to pull out.
Also, you describe them as "very close friends" - you were planning to spend a significant birthday with them. How much of an issue do you actually want to make this?
You are already living in another country to them, so if things turn sour it will not be terribly easy to repair. How much is their friendship worth to you? Are you prepared to sour many years worth of friendship over irritation around a short break that has gone tits up?
Yes, they should perhaps have been more considerate of your location and childcare commitments, but perhaps you should have been firmer on how problematic it would be at the time.
There is clearly already bad feeling on both sides around this. From the tone of the texts they have sent, it seems that perhaps they think you are being difficult - that you shouldn't have agreed to go if it was going to be problematic for you, and now that you've pulled out you're refusing to cover the costs.
It seems to me that your choice is either to pay the money and vow to be more assertive in the future about agreeing to plans that you're really not happy with, and this will all blow over and the friendship will remain.
Or you follow through on your anger and upset with them by witholding the money and things are never the same between the four of you again.