So, it's established that Stiglitz is well known in Germany and that the Nazis know he's running with the Basterds now. He isn't just a notorious serial killer; he's a serial killer who's still out there, killing officers, so it would be in the best interest of the Germans to remember his face. Especially if they're wearing a Gestapo uniform.
The Convenient Lapse in Logic:
Hugo Stiglitz is the most notorious member of the Basterds, so what do they do? They choose him for an undercover mission that absolutely depends on him not being recognized ... and he isn't.
The Basterds are recruited for a mission to infiltrate a film premiere attended by several high-ranking German officers, and Stiglitz is among those chosen to impersonate German officials, since he actually was one. Even if his face wasn't in the newspapers (it was), that's still a dumb plan, because all it takes is one person at the premiere who served alongside Stiglitz, was present during his incarceration or simply witnessed him scalping a fellow officer to ruin the entire carefully orchestrated plan.
It doesn't help that his name and theme song tend to magically appear anywhere he goes.
The dumbest part, however, is that the plan actually works (for a while). Before they can get to the premiere, some of the Basterds meet with a spy in a bar that happens to be full of German soldiers. So Hugo "The Whole German Army Has Heard of Him" Stiglitz is face to face with other soldiers, and not one of them, not even the Gestapo officer, so much as looks at him funny.
He started parting his hair on the other side. Threw the whole Gestapo off their game.
Eventually the Basterds are found out -- not because of Stiglitz, but simply because another of the undercover agents raised the wrong fingers when asking for beers. The Gestapo officer was so observant that he immediately picked up on the accent of another person talking across the room, but at no point did he recognize the well known Gestapo killer sitting literally next to him.
"Cheer up, friend! You look like you want to shoot me."