Like you say, they do appear to be having fun, which was rare for them at that point. Even the little lead in where they're mucking around with David Frost (singing "David Frost....David Frost") makes me grin. So yeah, I agree, it's Hey Jude.
That's why the picture below is my favourite of Paul & John - despite all the hassles that they were going through, they could still enjoy the music. And that's what we're left with - the music