Don't know if they ever played it live. I've found some setlist from early 1963;
Neither of them does mention I'll Be On My Way in their lists.
Strange to see how many cover versions they were still putting in their sets by April 1963, even though they had already released an LP plus two singles (and the songs for the third single had been written and recorded too - released on 11 April 1963).
So they had lots of original material plus a bunch of cover songs that they had recorded for their PPM album, yet they chose to perform other cover songs that never made it onto Beatles records at all (other than the BBC sessions they did).
Songs like Beautiful Dreamer, 3 Cool Cats, Hey Good Looking, Keep Your Hands Off My Baby, etc.
They obviously enjoyed playing them live but didn't feel they were suitable for a record release.
And it still doesn't answer why I'll Be On My Way was played on a BBC session when it had already been given to another artist.