Well, "twang" would be anathema to a McCartney sound, which if I understand the question correctly is more a function of the Hofner bass anyway.
I use the J-Station exclusively for recording bass at this point, despite having a bunch of good amps. The acoustic problems of bass amps in the studio makes using a modeler of some type almost a necessity. I could use the XLR outs on my Ampegs and SWR, but I prefer the J-Station for convenience.
Point one:
If you're not using J-Edit, you're probably not using 5% of what's available to you. Without J-Edit, the J-Station is nothing.
I usually use the Ampeg SVT amp model (#15 Rock) with any number of cabinet models. I think that's probably where you are going to find the sound you're looking for among the bass amp models. Pick the amp model, turn off effects and just scroll through the cabs that look attractive to you. It's amazing how much difference they make. Don't neglect to use guitar models as well. There's no reason not to try them.
Doink around with the compression a bit and EQ. My own secret weapon for big sound is to record in stereo with the "Analog Pong" effect set to one single iteration at full volume and tweak the delay somewhere between 10ms-35ms until I hit a sweet spot with no phasing problems. What this means is that you have the note and (say) 10ms later it's repeated once in the right channel and 20ms later repeated once in the left channel. It's undetectable as an effect. It just sounds really fat.
Here's a list I made of the actual J-Station models in the v.2.0 firmware. It may help you narrow down the search:
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Guitar & Bass Amp Models
00 J Crunch (based on a JM150 Millennium "crunch" setting)
01 J Solo (based on JM150 Millennium "solo" setting)
02 J Clean (based on JM150 Millennium "clean" setting)
03 Boutique (based on a Matchless
DC30)
04 Rectified (based on a Mesa Boogie dual rectifier)
05 Brit Stack (based on a Marshall JCM900)
06 Brit Class A (based on a '63
Vox AC30 top boost)
07 Blackface (based on a '65 Fender Twin Reverb)
08 Boat Back (a piezo acoustic guitar)
09 Flat Top (a dreadnought acoustic guitar)
10 Hot Rod (based on
a Mesa Boogie Mark II C)
11 Tweed (based on a '57 Fender Tweed Deluxe)
12 Blues (a dynamic blues setting)
13 Fuzz (a '60s fuzztone)
14 Modern (based on a Trace Elliot bass amp)
15 Rock (based on an Ampeg SVT bass amp)
More A1 (based on
a HiWatt Custom 50)
More A2 (based on a '78 Marshall master volume)
More A3 (based on a '81 Marshall JCM800 w/EL34s)
More A4 (based on a '72
Fender Bandmaster)
More A5 (based on a '65 Fender Bassman)
More A6 (based on a SWR Interstellar ODrive)
More A7 (based on a '83 Fender Concert Head)
More A8 (Direct - no modelling)
Cabinet Models
00 - No cabinet
01 - Brit 4X12 (Marshall 1960A w/75W Celestions)
02 - Johnson 4X12V (loaded w/Vintage 30 Celestions)
03 - Fane 4X12 (a HiWatt SE4123 w/Fanes)
04 - Johnson 2X12 (open back w/Vintage 30 Celestions)
05 - American 2X12 (a Fender Twin 2X12)
06 - Jennings Blue 2X12 (a '63 Vox AC30)
07 - Tweed 1X12 (a Fender Deluxe 1X12)
08 - Blonde 2X12 (a Bassman 2X12)
09 - Bass 4X10 w/Tweeter (an SWR 4X10 w/tweeter)
10 - Folded Horn 1X18 (an Acoustic 360)
11 - Flexi Bass (an Ampeg Portaflex)
12 - Green Back 4x12 (
a Marshall 1960B w/25W Celestion Greenbacks)
13 - Mega 1516 (a Peavey 1x15 and 2x8)
14 - Boutique 4x12 (a HT 4x12 w/Celestion V30s)
15 - '65 Tweed 1x12 (a Fender '65 Deluxe)
16 - Goliath 4x10 (an SWR Goliath)
17 - Ivy League 1x10 (a Fender Harvard)
18 - Bass Man 4x10 (a Fender Bassman)