Well, I always used to feel that way about "Oh Danny Boy," but my somewhat more financially astute/desperate partner amended my "It'll be a cold day in hell before we..." to "We'll only sing that one for a $20 tip." (By the way this was 1976, when $20 was a lot more than it is now.)

Much to my surprise, we got the tip. (Actually I kind of recall that the guy gave us $50, but I could be misremembering that.)

I developed a great fondness for Danny Boy that day!

So...I respectfully suggest a hefty "hokey song surcharge" for such requests. Usually the requester will demur -- whoopee, then you don't have to sing it! -- but if they say yes, the financial incentive can handsomely offset any momentary lapse in coolness.