Great tune, eh? Sounds like the type of thing Paul liked to do. Here's a great story about Smith's songwriting and the Beatles:
I'd been writing songs since I was a small boy, and in 1965 I wrote one with John Lennon in mind. They were coming to the end of the Help! LP and needed one more song. George Martin and I were in the control room waiting for them to make up their minds and I said 'I know they've heard all this before, but I happen to have a song in my pocket.' George said 'Get on the talkback and tell them.' But I was too nervous so George called down, 'Paul, can you come up? Norman's got a song for you.' Paul looked shocked. 'Really, Normal?' - that was one of their nicknames for me - 'Yes, really.' So we went across to Studio Three and I sat at the piano and bashed the song out. He said 'That's really good, I can hear John singing that!' So we got John up, he heard it, and said 'That's great. We'll do it.' Paul asked me to do a demo version, for them all to learn. Dick James, the music publisher, was there while all this was going on and before we went home that night he offered me £15,000 to buy the song outright. I couldn't talk but I looked across to George and his eyes were flicking up towards the ceiling, meaning 'ask for more'. So I said 'Look, Dick, I'll talk to you tomorrow about it.'
I did the demo but the next day The Beatles came in looking a little bit sheepish, long faces. 'Hello, Norm.' I thought, hmm, they're not as excited as me, what's wrong? Sure enough, Paul and John called me down to the studio and they said 'Look, we definitely like your song but we've realised that Ringo hasn't got a vocal on the LP, and he's got to have one. We'll do yours another time, eh?' That was my £15,000 gone in a flash. By the next LP they'd progressed so much that my song was never even considered again.
By the way, Norman wasn't working with The Beatles from about "Revolver" on....Geoff Emerick became their main engineer around the time "Paperback Writer" & "Rain" were done.