I will tell you what we did.
One was severe. All hay soaked, but my friend let him out to graze periodically. Unfortunately she did not keep the grasses mowed short. He was on meds (dexamethasone and albuterol). He did have a stall that was air conditioned because he developed anhidrosis (no sweating). Hated the stall, instead preferred to stand under a fan that had a mister on it. She passed away a couple of years ago and left her two horses to me. By then he was pretty bad: full blown anhidrosis, not eating, breathing labored no matter what meds we used. I made the decision to euthanize him when he stopped wanting to eat.
My horse was diagnosed this summer. I handled it differently. I used Dex at 10cc for a few days, then 5cc for a few days to get everything under control and then tapered back, using only 2ccs a day to maintain. I built him his own paddock off the main field. First it was a dry lot, but the dust aggravated him so I opened it up larger including about an acre of grass, but I kept it cut short weekly so no seed heads came up (which is what many are allergic to). He got Senior feed, NO hay. He actually lost the huge 'hay belly' that he had when on pasture. He was always a very easy keeper, getting a handful of feed just to come up. He looked better and more fit. He was doing really good, but I knew it would get worse if he stayed here. Got him moved into a home further north west and he is totally off the meds and doing great.
A friend had one that was so severe that it looked like a walking skeleton. It was doing poorly in it's current living situation (on pasture). The lady wanted to do right by the horse and had tried things to no avail. She looked into boarding with my friend and with only a sand lot for turnout. It's working for this horse. He has turnout for some hours a day in a round pen, otherwise stalled. The lady is riding him all over the place though and his weight is amazing. He gets NO hay either. He is only fed a complete Senior feed, four times a day, which has no dust. He is not on meds anymore. He has had no flairups, but is never allowed to eat grass or hay...
Roundbales are the worst for heave horses. You can take parts of it off (keeping the rest under cover) and then soak it, but honestly going off the hay and onto a complete senior feed multiple times a day does really work as they aren't exposed to the seed and dust of the hay.
I hope you can get it under control. It takes more management to keep them comfortable.