Main Street Songs, Inc. was the Miller's BMI imprint so the '44 copyright may just be back office paperwork for the movie.

The play came out in December of 1943. So if any part of the Moss Hart back story is real, the 'based on the true story' AAF cadences should already be in pretty good circulation.

fwiw Seeger &co.: Bob Miller Songs, Inc. published Joe Hill!

fwiw Gitz Rice: Bob Miller and his Hinky Dinkers, feat. Charlotte Miller. And at least one copyright on Hinky, Dinky, Parley Voo (1936) Wouldn't be too surprised if it's the same one or similar in Winged Victory.