Why not Wendell Willkie?
He was a registered Democrat until he decided to run for the GOP nomination.
He supported most of FDR's big projects, and was an internationalist, so he might be acceptable.
However to get Willkie as the DEM nominee, we need either the GOP, led by Taft, Vandeberg, and Dewey to say "NO we don't want you"
or
Have FDR suffer a stroke, or some other health debilitation right after the 1938 midterm election that makes a run in 1940 impossible. Perhaps in that position, FDR might want a young, charismatic, forward looking man to continue his legacy.
I'm not sure on running mates, but no Willkie means either taft, or Vandeberg are the GOP nominee.
Wishing you well, his majesty,
The Scandinavian Emperor