D.B.
As usual, you are far more open to other points of view than many people are. Including your ex.
If your son had asked for this, begged for it or even agreed to it, there'd be some value to "stick it out" and "talk to the coach about ways not to be ready to vomit" (dialing it back, changing what he eats beforehand, warming up differently, etc.). And I do think kids shouldn't instantly quit what they ask parents to pay for.
But that's not what happened here. Your son, at 13, didn't have any say at all, and you weren't consulted. So your ex, whom we all know is difficult, decided he knew what was best and he wanted to be in control. And he still wants to be in control. Maybe he's mad at you, maybe he's mad at your son, maybe he thinks his value as a man or a parent is based on how tough his teenager is. You did urge your son to consider talking to the coach etc. but maybe the coach isn't approachable or maybe your son is already so pissed and feeling used/abused by his father that he's in no mood to give him any slack.
I think I'd brainstorm a little with your son about ways to approach his father - or perhaps write a letter - and I think it might be an good exercise in growth and maturity for your son to talk to the coach, even if it's just to explain that "This program isn't for me" and either say why (to help the coach perhaps improve if other kids have this problem) or at least learn to exit gracefully and politely. But other than that, I'd say you're totally on the right track here. Your ex's attitude is what drives kids away from these programs anyway. And it's what drove you away from him, if I recall!