We have the same issue in our house: Most hay is cut when pollen is still enclosed inside of the seed pods because it is the most nutrient rich point on the grasses lifespan. When dried it permeates the hay. Unfortunately 3/6ths of the member’s in our family are asthmatic to hay pollen. When we chose to get rabbits (because they also found in necessary to be allergic to dogs and cats ?) I determined ahead of time to harvest and dry grass during the summer months when it was not in pollen (right before it starts to get seed heads). Thankfully our neighbor has a ditch of great wide leaf grass that is too mucky to mow and they let me tediously hand cut a years worth. I must have looked crazy, but our buns are happily munching the spoils of my labor and no one is having any kind of allergic reactions.
I am not sure if you live in an area where you have a yard plot you could plant and let grow, or nearby field or something of that sort, but could this be a possible option? If it was something you are interested in and need any tips on properly curing the hay so it won’t mold later, I will be glad to give you some tips.