In the end, I really do think the socialist label would have doomed him.
Hillary's Republican courtship strategy was a good idea in theory, but it failed because everyone realized she had no common ground with them aside from some areas of foreign policy. The fact of the matter is that Hillary is not a moderate, she is a partisan and proud of it. It didn't matter that she regularly invoked God on the trail or that she promised to have an "open-door" policy toward republicans talking to her when she was in office. Everyone knew that she was a partisan. From speaking in favor of partial birth abortion to having Obama come right out and demonize people who were planning to vote Hillary/Republican Senator, she made it very clear. If you want to court the other party, you need to give them a reason why. And "I'm not Trump and I might do what you want on foreign policy sometimes. Maybe. AND RINO BLOOMBERG SUPPORTS ME!!!!!!" is not an actual reason.
The way 'Socialist Obama's' middle name being Hussein and his connections to Reverend Wright, etc. doomed him? Most Republicans believe Obama is a muslim, you think a muslim is electable but a socialist (let's be a honest, a social democrat) is not? If there was anything Bernie was known for during the primary season it was him calling himself a socialist.
Going by your logic, no dem would be able to court republicans in the first place. This speaks to political polarization and hence the favorables for the most well-known candidates like Jeb, Hillary, Trump and (Cruz at the end) being what they were, leading to my second point. When people learn more about the actual positions of those candidates they go down due to polarization.
You can court the other party if you give them a real reason to come over to you. For instance the Trump Democrats in the rust belt came over because Trump made it part of his central message to tear up NAFTA to bring jobs back, a clear contradiction to the usual republican position of "free trade is amazing", and exactly what the rust belt wanted to hear.
