La Foiette a Republican candidate for Presdient

Not likely. Even TR couldn't oust Taft in a primary fight, and the party leadership didn't like Taft. What's likely is more like the Dems in 1952 IRL: LaF challenges Harding in the primary and does some damage, Harding drops out, and the leadership taps someone at the convention (maybe Coolidge, maybe someone else) to derail him.
 
Is that plausable, who else was on the VP list?

Irvine Lenroot, a fellow Wisconsin Senator and LaFolette protégé, was actually the favorite to win the VP nomination but was ousted by a dissenter delegate who entered Coolidge's name. I tried to make a TL once about that.
 
Irvine Lenroot, a fellow Wisconsin Senator and LaFolette protégé, was actually the favorite to win the VP nomination but was ousted by a dissenter delegate who entered Coolidge's name. I tried to make a TL once about that.

Lenroot, as you said, was a former LaFollette protege, but by 1920 the two had had a number of serious fallouts, to the point that LaFollette had tried to prevent Lenroot's election to the Senate (the Progressives seem to have felt that Lenroot, in switching his support to the moderates in the state, had betrayed the faction. They worked hard to engineer his defeat to former-governor in 26).
I think the big problem for Lafollette, by 1920, is that he'd simply burned too many bridges with the Party, which by that point was moving more conservative. His best chance to gaining the presidency was in either 1908 or 1912. In both of those instances, he almost is required to gain the support if Roosevelt. Unfortunately TR and LaFollette both suffered from a chronic case of testosterone poisoning in relation to one another.
 
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