Terrible cat allergy question?

If you wash it alone it shouldn't be a problem. You could even do an empty cycle after to make sure the washer is clean. I suppose drying it would be more worrisome for me. If there's a leak anywhere leading to your vent. contaminated air could leak into your walls, basement..lol :) Maybe you could air dry it. If you really want to be safe, put it in a garbage bag or something and take it to the laundromat.