Of course different people have different severity of nosebleeds. When I was a boy I had fairly frequent nosebleeds. My Dad took me to a specialist for that and they cauterized my nose with a silver nitrate solution (they still do that by the way). I also learned a few things to minimize the collateral damage from the doctor: Stuff a little wad of tissue up the nostril that’s bleeding and take it easy for a while - Don’t move around a lot until the bleeding has stopped. And always carry Kleenex or a cotton handkerchief to control the blood. And bloodstains come out with cold water when they are fresh - Just soak the stain in cold water, assuming it’s washable and not dry clean only. My handkerchiefs were sometimes a bloody mess but the cold water soak and rinse when I got home followed by a wash always took care of the stains. It comes out best when it’s treated before it dries. Also, ordinary drugstore hydrogen peroxide can remove bloodstains - but you have to be careful and promptly rinse it out, you don’t want peroxide sitting on fabric a long time as it can bleach the fabric. Or you could take the costume to a good dry cleaners and explain what happened and hope they can get the bloodstains out. That has some chance of taking care of the problem at a lower cost.

You should flip it around and say you had an allergic reaction to their improperly cleaned costume and threaten legal action.