What about religion? Religion is, strictly speaking, mutable. But, for me at least, saying "no Muslims" or "no Sikhs" is the same level of badness as saying "no blacks".

Religion is a tricky one.... It is mutable, but most people (a) don't change their religion in their lifetime and (b) use religion as a proxy signal for social identity, including many things that aren't mutable.

In practice: I'd take the bet that a white person of Muslim faith could get around a "no Muslims" rule most of the time. "No Muslims", then, can be a proxy for "no middle-Eastern looking folks".