And if you do "Get Back," or "Hey Jude," or "Bridge over Troubled Water," aren't you really cheating? I mean, Elvis may have recorded them, but not many of us really think of them as Elvis songs.
So then is it only "the top forty hits" that count? Maybe, but man, does that eliminate lots of great songs, and maybe the best of Elvis' work was not on the top forty. He sang Country, Gospel, Blues, Rock & Roll, and Schmaltz and Dreck, and just about everything in between too.
And if you sing an Elvis song, you find yourself trying to sing like Elvis. I think this may explain the Elvis impersonator phenomenon. You not only want to sing songs he sang but you want to sing them like he sang them. Elvis had such a big, distinctive (some might say bombastic) style that sometimes totally overwhelmed songs he sang. This was both good and bad. Elvis became the King, and everything he touched became Elvis, and just like King Midas this kind of became a tragic flaw too.
If you find yourself in Las Vegas, big, bloated, fat, drug-addled and sick, and all the adoring fans are still flinging their underpants at you, and you are forgetting lyrics, and stumbling around onstage, and everyone is still telling you that you are "the King" that can do no wrong… well, do you kind of find the whole thing sort of hollow and unfulfilling?
Anyway, not sure how to resolve the "Elvis Song" question… here's Elvis in rehearsal doing the Beatles "Hey Jude"… "let it out and shove it in!" Hah!