What happens if John H. Bankhead was FDR's Vice President in 1944?
AFAIK, Bankhead was not even considered. The most likely southern VP candidate was James Byrnes of South Carolina: former Representative (1910-1924), Senator (1930-1941), Supreme Court Justice (1941-1942),
"assistant President" (FDR's war economy czar) (1942-1944).
FDR in his charmingly duplicitous way let Byrnes think he would be the nominee. Truman went to the convention expecting to make the nominating speech for Byrnes.
If Byrnes is VP - it may alienate some black votes in 1944, but it won't affect the election.
He becomes President in 1945 instead of Sec of State as in OTL. I don't see his foreign policy being much different from Truman's.
Where the rubber meets the road is 1948. Byrnes will not desegregate the Armed Forces. There will be no civil rights plank in the Democrat platform, and obviously no Dixiecrat ticket. That means a much stronger campaign by Henry Wallace. (OTL, a lot of liberals who had doubts about Truman were reassured by the enmity of the Dixiecrats. ATL, they'll go with Wallace.)
That probably means that the Republicans adopt and publicize a strong civil rights plank, and win back a lot of the black votes that were shifting Democrat. (Those votes were more important by 1948, with the Great Migration in progress.)
Between extra black votes, and additional liberals splitting to Wallace, Dewey will win. The Republicans will be the civil rights party. Warren will be VP, not Chief Justice.