Two main points. The venue hired you to do what they know you CAN do, unless you actually billed yourself as a DJ and an "I can play ANY tune, any time" novelty act. So a request FAR from your normal repertoire can quite easily be deflected with an "I'm sorry, I don't play that song. Here's another one you might like, and I'll play your request on the break from my DJ collection" (not many here can sing Sting anyway!).

But also, the venue is hiring you to provide the music they thinks best suits the majority of their customers. So, if you are at an upscale restaurant or country club, and some kid eating there with his parents comes up and wants the latest Mettalica, you don't for one minute think you are obliged to play it, do you?

In the original case, I would have said "sorry, I don't do that one, but do you like Fields of Gold, or Every Breath You Take, or Fragile?", Sting songs that are more appropriate to the venue. If not needed, at least you have shown a willingness to meet the requester half-way, and they go back to their seat happy. Or if they are still unsatisfied, you have at least TRIED, and you aren't a juke-box, no matter what the stupid brat thinks you should be!

And the venue owner will be grateful you didn't spoil the mood of his place!