Personally, I would neither show up at the wrong desk (in this case, business class) nor even ask about using it.

That's because I know how much I resent others using facilities to which they're not entitled, and so I don't do it myself. If I were to travel in Y (which, admittedly I tend not to do for medium or long-haul stuff) and with a carrier where I have no status, then I would happily adjust my expectations accordingly.

I'm very conscious that some folk (perhaps even many ...?) would be inclined to say what the heck, this is no big deal. But I happen to believe that this is precisely how priority check-in and boarding zones can become unnecessarily crowded - at times, so much so, that their whole raison d'être can be defeated.