Hmmm, might be tricky. I can see him as a Republican in a TL where Progressives control the Republicans from 1912 onward, but I'm not sure how easy that is after then.
*maybe* Dewey in '44. let's say FDR is very sick but doesn't die soon before the election, and some other things happen that make Dewey look very presidential. Not sure what but ISTR someone did a TL like that once but it didn't get much further than that.
The only question is, are the Southerners powerful enough to put Thurmond in power in '48? If Dewey is the one to integrate the military, and then the Dixiecrats control it so Dewey wins a 2nd term.
Then, hwoever, you'd have to get LBJ so disgusted witht he waythe democrats have gone that he shifts to the Republicans. Would he?
There could be a way - suppose some shady stuff keeps him from winning election to the Senate for the first time. He might run as an independent and, if popular enough, win. then, as he tries to work with the Democrats,, their attitude so frustrates him that he drifts closer to the Repubican camp. Eventually he's a compromise nominee witht he Progressive GOP against the very conservative Democrats.
Okay, all of that *might* work, but how to actually put it into practice, I have no clue. Johnson was prideful enough, though, that if all the other stuff happens with his political career, he might just snub the Democrats and go to the republicans. But, whether it can, I don't know.