I can't think of any Hart winners with playoff careers nearly as bad as Bobrovsky or Carey.
Off the top of my head:
Nels Stewart was terrible in the playoffs after his rookie year. But he was best playoff performer on the Cup winner his rookie year.
Roy Worters never won a full playoff series in his career, but I think that says at least as much about his team as it does about him.
Milt Schmidt's playoff stats aren't very good, but they are better than these goalies, plus he was excellent defensively, including in the playoffs.
Andy Bathgate might be the best answer here, but his playoff stats are merely bad, not pathetic, and he did have the detriment of basically zero team depth for most of his career.
Joe Thornton's had some decent moments in an overall weak playoff career, more than you can say of those two goalies.
disagree on Schmidt, for the time his numbers are good.
he led the playoffs in scoring in '41 with 11 in 11 (certainly a Smythe win if it existed then) and followed it up with 8 in 10 in '46
Dont forget Schmidt missed 4 years of his prime (ages 23-26) because he was busy fighting Nazis.
The war years ended what could've been a Bruins dynasty in the 40s. They won in 39 and 41, then lost their entire top line and the best goalie in hockey to the war effort in Schmidt, Dumart, Bauer and Brimsek in '42. Thats 4 hall of famers gone at once, and all in their primes ages 22, 24, 25, 26.
They had a 20-12-5 record at the time and looked on their way to repeating.
Basically, if not for the War Schmidt adds to his 2 Cups and has 2-4 more deep playoff runs as the best player on the team on his resume