Certain sins are so bad, so unnatural, that they shouldn't even be named, let alone tolerated in public. The fact that they are manifestly ɧoɱosɛҳųαƖ means they are flaunting it, publicly putting it on display.
Bear in mind that the ɧoɱosɛҳųαƖ cannot reproduce according to nature. Their principal means of "reproduction", then, is to spread their vice to others in society. Among the various means they employ is to put it on display in public, giving the false impression that it is "normal", and by intimidating the Christian into tacitly admitting (by his tolerating them) that their behavior is "acceptable."
Refusing to serve them is absolutely warranted. They must be clearly, manifestly of that "persuasion", of course. It's not enough to simply suspect.
If the laws of your locale are such that it is impossible to refuse service without incurring the wrath of the state, then other means of dissuasion can and should be employed.