So, my horse has heaves?

It does sound like he's reacted to something in the round bale of hay. Generally with heaves you want out as much as possible as that tends to be the most dust free. You may need to feed soaked hay but I would try the bales from different suppliers and see if that helps. You're going to have to be really careful about the quality of hay he gets and use a dust free bedding so shavings of some sort, definately not straw.