I still don't understand. Why do you attend a session in a public place if not because there's an audience there? If you don't want/don't care about/are bothered by the public, why don't you just have your friends over and play in the kitchen? Trying to pretend you're playing music in pub for any reason other than to play music in public is kind of silly.

Why on earth would anyone get annoyed because a member of your audience (yes they are, whether you admit it or not) asks for a request? You can say, respectfully, "Sorry, we don't know that tune" or "Sure, we'd love to play it" or "this is an Irish session and we need to stick to Irish music".