I'm the man on the flaming pie
'I'm the man on the flaming pie' says Paul, which could be read in a number of different ways. If he's speaking as John, then you have the implication that Lennon came up with the name 'The Beatles' himself - in a sense, John was the man in his own vision. If Paul is speaking as himself, however, then you may have a reference to John's famous faux-confession 'the Walrus is Paul' from the song 'Glass Onion'. Perhaps Paul is correcting him? He's not the Walrus, oh no, but the man on the Flaming Pie.
Or, more likely, Paul is simply taking up John's mantle, and daring people to find meaning in nonsense...