The Beatles released "From Me To You" as a single (with "Thank You Girl" as the B-side) on 11 April 1963. In the middle 8 of "From Me To You" are the lyrics "I got lips that long to kiss you And keep you satisfied", the last three words of which are shared with the song "I'll Keep You Satisfied". It is uncertain when McCartney composed "I'll Keep You Satisfied" (whether it was before or after "From Me To You"), but it would seem most logical that the line in "From Me To You" prompted the birth of "I'll Keep You Satisfied", although I have never encountered a quote from Paul to support this notion.
Regardless of chronology, the tune was given to Billy J. Kramer and the Dakotas, who recorded the song at Abbey Road Studios on 14 October 1963 under the supervision of producer George Martin. (John Lennon was present at the recording, although why is uncertain because it was primarily Paul's tune.) Their recording was then released the following November 1.
The Beatles never recorded "I'll Keep You Satisfied", not even in demo form.
This blog is a workshop for developing my analyses of The Beatles' music.