I think so, because the Beatles have covered other Little Richard tunes and he was one of their personal favorites. But to find a record collection with that cover song by them seems like a difficult search. But it pops up on the internet. Have you had any luck yet?
At one of the other pages, someone said they did it for the BBC, but I don't know about that. My guess would be the Get back sessions. Either way, I'm near definite that they performed it - even recorded it; I'm just not entirely sure where or when.