Have someone start taking videos of every shot. That will get you used to being watched. Have them tell you they are putting it on youtube as soon as the round is over.
If the tailgater is not actually making any noise or distractions, it is all in your head, and the most likely issue is you being self conscious about others watching you throw/putt, etc.
For anyone who plays tournaments, especially starting out, this can be a real thing--but having someone video you can help you focus. Its all about focus. I'm admittedly not great at it, but I know what the problem is anyway.
In fact, that tailgater can help you work on your focus, for free. Let 'em stare! Also, any pressure you feel about holding them up--that's also in your head. You offered (maybe more than once), they declined. Play your round at your speed.