I would have a chat to try to figure out what exactly is going on. I would say that the heat and being tired are probably playing into it. Not everyone copes well. I'm local and I have seen SO MANY arguing families, especially right now in the Summer. So it's not just yours. People are hot, tired, getting up early, going all day, dehydrated, half of them are constipated from being on a different schedule/diet, different expectations within families. Parents constantly barking at kids or threatening them. Kids constantly whining or crying. Couples arguing. All the time.
How would he do roaming the park on his own? I have teens and they often roam alone and I give them the ability to mobile order and they enjoy that freedom.
14 is often a harder age than 16- hormones and all of that. The back and forth and threatening of sending him home and feeling like he's not meeting your expectations probably isn't helping, either. Because in all reality, I'm going to guess that you're feeling stressed because of the cost and that the vacation isn't living up to your own expectations. And you and your husband are probably extra stressed because of the heat and the go, go, go.