Both of them just seem so darned appropriate.
It's WEE-ner.
Wiener is the German/Yiddish word for "Viennese" somebody from Vienna. Prounouced VEENER. Wienerwurst= Viennese Sausage. Wiener Schnitzel=Viennese Veal Cutlet. So if the guy's name is Weiner, it's Veener in German or Yiddish, and Weener in American.
To judge from the dweeb's appearance and the spelling, I would pronounce the Weener's name "Whiner."