If you are going to take this in front of a judge, you're going to have a get your story more straighter than you have here, because what you've written makes no sense.
Firstly the trip was £1,400 including flights of £150 and an additional flight of £200 to visit family. The latter making no sense at all. A different holiday, a second holiday within the other holiday?
Then it changes to flights being £150 and he owing £850, so £1000 all together, that was the deal you agreed, but it's really £1,600 all together (you deducted the £200 he agreed to pay).
You later mention the holiday is an all inclusive, so if the flight is £150, that means that his share of the hotel is £1,350. That's 1/5 of the cost (you, your OH, your DD and your DS), so you are saying that the hotel only cost £1,350x5? Where are you going that the flights are so cheap but the hotel ridiculously priced?
Finally, if the significant sum is mainly due to the hotel, why can only he go? We know that flights are none transferable in most instances, but I've never heard of a hotel where you can't change booking names.
I'm wondering if it is a cruise. If so, the cost of one room for one person rather than 2 is usually the same cost, or just about, so it wouldn't have cost any less if he'd never agreed to go in the first place. I also believe that cruises will allow a change of name.
I'm wondering if that is the reason the family is refusing to pay, because his not going, bare the flights is not incurring any additional costs to you. In this case, good luck presenting your case in front of the judge!