Como, previously with Decca, was with RCA Victor by 1943 and they were rather territorial about licensing recordings outside of that label. While he was allowed to sing in MGM's Words & Music, those vocals were not permitted, contractually to be included on any early MGM soundtrack album, nor was Metro interested in licensing them to RCA Victor. Como's label did allow him to 're-record a handful of those songs for their label. But there's just something about the MGM style that's lost in those re-issues.Yes, Peterson is on the various LPs and is credited. If I recall, it's the last cut on at least one of the albums of it I have.
You're right, SInatra is nowhere to be heard on the LPs, just as Perry Como isn't on the Words and Music soundtrack LPs.
The 'Composer's Collection' LD release of Till The Clouds Roll By ,Words and Music and Deep in My Heart from 1991, contained isolated 'audio only' recordings of all of Como's cuts, flubs and deleted songs in addition to the songs he recorded that stayed in the movie. One outstanding exclusion that did not survive the movie cut, but was included as a extra here, "You're Nearer" - a gorgeous ballad in which Como positively tears the heart out. Good stuff!