Yes, Sir Paul offered up his - if you want it, here it is "Come and Get it" - tune in an effort to jump start their career. BF was also signed to the Apple recording label and was well known by the Fab Four. Many comparisons were made at the time which has been rumored to have both helped and hurt (BF).The Beatles also used BF for some backing tracks (guitars and vocals) on Ringo solo "It Don't Come Easy" and George's "All Things Must Pass". They even were backing bandmates on various post-Beatle solo tours.
Hope the licensing works out.... I think it'd be great fun and a neat study to cover one of their songs!