Visited Altons weekend just gone. I can't wear a...

Visited Altons weekend just gone. I can't wear a facemask for medical reasons and usually shops ask me to wear a mask and I explain I am mask exempt and it's fine. This is the process outlined on the government website. However, Altons' door staff were extremely rude and demanded I wear a lanyard or the like even though this is not the government's expectation. They stated they had the right to demand this as it is private property and they can request what they like. Does this mean they are above the law and can discriminate on any basis they wish? Racial or disability for example. I don't think so. Also, I think this aggressive approach to mask wearing would be very upsetting for some people. For example, someone who has suffered some form of abuse. Really disappointing from a shop I used to love.