Purple I'd say WN was being unreasonable and unkind because he's refusing to even discuss a compromise. Ok, so he's booked tickets to something, and it's his weekend. But, is there any way that you could attend some of the party, visit GP, and return them in time to go off with WN ?
He could say to you if GP is so ill, why aren't you visiting any way, why is it only in conjunction with the party ? If so, why not visit GP on another day, not his weekend ? I'm not saying this is what you should do for a moment, just how he may see it.
Also, like you say, things are going to crop up. They won't wait 'till the right weekend. For this reason I think it is significant what the tickets are for. If it's meet and greet the England football team with signed shirts, then maybe the party could be missed. If it's cinema tickets, well...
If you turned the situation round, what would you do if you had tickets for something on your weekend, then he presented the party/GP issue ? Assuming your Dc felt the same about the party and GP in question, could you ask him to consider doing whatever you would do in that situation?
Ultimately, like you say, it's about what the dc want. If he won't even discuss a compromise I'd be tempted to say this is what the boys want, party and GP visit. If you tell me when and where the event is we'll work around it. If you won't, I have to assume it doesn't exist, could easily be rearranged or worked around, and therefore you are simply being difficult. If this is the case, the dc come first, so they'll be ready for pick up at xx (time after party and GP visit).