Hay should not be eliminated from a bun’s diet, because hay is prerty much vital for bun dental- and GI health. There are some things you can do though. I am allergic to timothy hay, and timothy seems to be the type of grass that most allergic people react to. I try not to have timothy hay in my bedroom (or I dry cough all night long). I tolerate orchard and oat hay very well. Rabbits do well on any type of grass hay, the dont absolutely have to have timothy, it’s just a very common type of hay.
It’s also important to keep hay dust down to a minimum. Allergic people have sensitive airways, and particles that are not allergenic can still irritate their airways mechanically. (During pollen season the airways are already under a lot of strain, and likely to respond “disproportionately” to the mechanical stress from dust particles). If I get hay that I think is too dusty, I pour the sack out on a bedsheet (I do this outdoors), then put all the hay back in a bag minus the dust/chaff that has collected on the bedsheet. (The bedsheet obv has to be shaken out outdoors and then laundered).
You can also spray some water on the hay when you serve it (not too much, the hay mustn’t get soaked) in order to bind some of the dust.
There are compressed hay cubes and “mini bales” that can substitue some long stranded hay (but it’s not recommended to substitute all hay with cubes). They’re obviously a lot less messy than loose hay and some buns really like them, whereas other buns will more or less ignore them. Just make sure they cubes/mini bales are not all alfalfa hay, because alfalfa is too high in calcium and protein for healthy adult buns.