It's real!!! The only viewable version I can find is here (site might be dodgy, use at your own discretion): https://avhim.ru.net/v/176611

It was not a video as such, but a performance for German music TV show called Bananas that ran from 1981 to 1984. It's as you remember, George in matching denim jacket and jeans dancing and a woman in a fat suit dancing behind. This link has some still images from the video: http://www.clip-vip.com/v13/dbabas2/16.jpg

After further searching, it has been released once on DVD as part of a collection called "Das Beste aus Bananas" (The Best of Bananas) in 2013. There's an Amazon listing for it here and if you look at the photos it shows the track listing on the back - Wham! Bad Boys is on disc 2 as track 16.