I'm curious. Do the women who protest against covering their hair at monasteries also refuse to wear a skirt?
Many monasteries prohibit women from entering if they are wearing pants.
I doubt that very many women would go to a monastery and refuse to meet the clothing requirements of a monastery. If I really had a problem with covering my head or wearing a long skirt or dress, I wouldn't go to a monastery. I don't wear a headcovering, but as you said, I would do it if my priest wanted me (and the other women) to. When I visit another parish, when I email the priest to let him know that I am coming, I ask about headcoverings. If it was the policy to wear a headcovering, I would bring one and wear it.