At most, I could see him as a featured artist credited on a single release (like the great Billy Preston on "Get Back" - I'm pretty sure this is the only time an artist from outside the band was given any kind billing for their performance with the group, and Preston was somebody they'd been friends with for most of the 1960s).
However, Billy Joel didn't have a hit until 1973 and even then didn't break through until 1977, having spent the intervening four years hiding out in Los Angeles while his lawyers tried to get him out of his rubbish management contract. He'd done sessions in New York and he'd recorded one album with the Hassles by the end of 1967; maybe if that album had turned out to be a monster he could've done something with the Beatles, but the time had more or less passed for the music they were playing by the time they showed up. The balance of power had shifted away from London and New York and now tilted toward Los Angeles and San Francisco; everybody was into psychedelic bullshit after Monterey.
In any case, the Beatles without John, Paul, George or Ringo wouldn't be the Beatles. If Paul McCartney and Ringo Starr teamed up with, I don't know, Noel Gallagher and Paul Weller and toured as the Beatles today, it wouldn't be the Beatles at all, would it? It's because the Beatles was and is John, Paul, George and Ringo.