GOP wins Minnesota governor's race 1962--does it change LBJ's 1964 choice of a running mate?

Suppose the Republicans won the extremely close 1962 Minnesota gubernatorial race. https://en.wikipedia.org/wiki/Minnesota_gubernatorial_election,_1962 The governor was for the first time elected for four years rather than two. This means that if LBJ were to choose as his running mate one of the two senators from Minnesota--either Hubert Humphrey (who was of course chosen in OTL) or Eugene McCarthy (who seems to have been considered for a while)--and the Democratic national ticket won, it would be Republican governor Elmer Andersen who would decide on Humphrey's or McCarthy's replacement in the Senate until the next election.

Would this fact dissuade LBJ from choosing Humphrey or McCarthy as his running mate? You might say he had more important considerations than a single seat in the Senate. But whereas in the House, a single vote very rarely makes a difference, it can do so in the Senate...
 
McCarthy was up for reelection in 1964. Considering LBJ's coattails, any Democrat nominee for McCarthy's Senate seat would very likely win (Walter Mondale, wink wink).
 
McCarthy was up for reelection in 1964. Considering LBJ's coattails, any Democrat nominee for McCarthy's Senate seat would very likely win (Walter Mondale, wink wink).

Ah, I had forgotten that. Would it, incidentally, have been legally possible for McCarthy to run both for VP and Senator a la LBJ in 1960, Bentsen in 1988 , Lieberman in 2000, and Biden in 2008? (Not that I think he'd do so, even if it were legally possible, because the Democratic national ticket looked like a sure winner in the summer of 1964, which was not true of any of the other years--not even 2008 before the autumn financial crisis.)
 
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