There's heaps of different rugs you can get.
During winter on my arab I use a cotton one and in summer I use a fly rug. There's also Jute rugs and that will act as a grooming tool as well. It makes them super shiny and removes loose hairs etc. I think that might benifit you a lot with your allergies and asthma not having to spent so much time grooming.
It's an Australia website, but that's what I'm talking about anyway Horse Rugs, Jute: Buy online at Horse Supplies Direct