I was able to get very close to his sound on my Hofner Club through a Portaflex, but I also used to get a workable sound on my Epiphone Rivoli.
Of course, you have to start with a set of flatwounds. I'd set my EQ on my bass and amp to get full lows, full lower mids, and highs cut to almost zero. On my Rivoli, I would use a small piece of sponge near the bridge to slightly dampen sustain. Also, probably because of the low-powered amps of the time, you might want to add a little compression and a touch of overdrive to get close to his tone. YMMV.
Most importantly, of course, as frono mentions, technique plays a huge role, especially when choosing to use pick or fingers and muting with the picking/plucking hand. This is a song-by-song decision, and deciphering whether Macca was using his Hofner or a Fender Jazz.