I've done lots of long distance open water fly (up to 2.4 miles at least twice, once with a full sleeved wetsuit on which was tiring as hell) and yes, overdoing the amplitude will mess you up. You need to learn to barely kick, and then that will keep you a lot more level. It's funny, because not kicking much is actually faster when going longer. One way to train yourself is with a pull bouy to keep your legs up, but then add the zoomer shortie fins so you get some propulsion and don't stall out.

Another thing is to slow down your turnover so you quit trying so hard. Then you'll need less oxygen. And then you can breathe every stroke, because there's a longer interval between strokes anyway. And then you're not having your issue in the first place. And you're also going faster because your amplitude is shallower. It all works out if you just slowwwww down. But that applies to long sets, not fast short ones.

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