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Stores do this already. Signs which read "Restrooms for customers only" are not uncommon.

That's not a "class" of customer, that's a status of being/not being a customer.

Class of customer, in the sense of the question, is a visual attribute of a customer making them unwelcome due the low probability of them being profitable for the store.

I can imagine businesses simply prohibiting the use of the toilets to people who don't buy things (“restrooms for customer use only”), which is a closer analogy: it's not about a retail store where you are browsing and may or may not buy, it's a place where you “buy” or not (by accepting ads) when you enter, and your access to other facilities is conditioned on that decision.

It's not unusual to see stores which have signs on their washrooms stating that they are "for customer use only".

If they banned people who walked in proclaiming that they had no money and no intent to buy anything - sure. Doesn't seem that strange for a shop.