This is a great question, I know we've discussed this before but I want to offer some suggestions here as well. I know some of us made some suggestions that were pretty good so I'd also recommend searching the archives, I can't remember where those discussions are posted!
I'll say similar recommendations on various sources for "getting that tone" assume everyone has the exact equipment as on the original tracks, guitars/amps/effects, etc. I'd rather go with the tools available, or more easily and readily available.
The bass itself: Do you have a Fender style bass? Carole and Ray Pohlman had Fender P-Basses on most of those tracks, Joe Osborn had a Jazz Bass. I mention them because all three were known for being pick players, that was a key part of the sound. More plucky than, say, Jamerson on Motown tracks.
If you have a Fender style bass, start by having the tone knobs all the way up. I also always recorded bass with the volume at full - in other words, every control set to "10", full out. I think the tone of the bass comes out better that way.
Use a heavy pick. Anything lower than heavy doesn't work as well.
Palm mute. This is another key - although Carole for one would put masking tape at the bridge to mute, but sometimes for different reasons. I use a piece of foam which I cut to size and fit under the strings at the bridge, barely touching the strings. I do this to cut overtones and other noises to avoid when recording direct. Make the foam thicker and more of a deliberate "mute" for the extreme effect. But I would say if you're not palm muting with your picking hand resting on the bridge, it's not going to get the sound.
Amps: Do you have an actual bass amp, vintage/modern? Or do you run through a simulator like a Line 6 Pod type device? Or an amp modeler either on a DAW plug in or an iPhone/iRig type of setup?
If you have a modeling plug in or iRig type of setup, that's the easiest way to test this out.
Carole for one would run bass through a Fender tube guitar amp, I believe it was a Super Reverb. If you have an amp modeling program, try dialing up a Fender Twin guitar amp and run your bass through it. No real need to tweak the tone eq that much, just like a real Fender I'd dial in anywhere around 7-8 on bass, mid, and treble, and if needed you can roll off the treble knob and boost the bass.
Definitely add reverb, enough to be noticeable. On the actual amp, I'd start around 5 and add or roll off as necessary. Reverb on bass? It sounds crazy now, but try it out. That reverb helps give some of the "resonance" you mentioned, as the picked, muted note has no sustain but the reverb keeps the sound ringing after the initial attack.
Then for an added layer which is on those BB's tracks, put on a slapback tape echo effect. Set it like a rockabilly slapback, keep the delayed note close to the original, and adjust levels to suit. In this case, a little less volume on the echo'ed signal combined with the reverb adds even more resonance and depth to what is ultimately a dead note going in.
If you're trying to get the sound through a standard, solid state bass amp it can be do-able, but you really have to focus more on the guitar-like qualities and set the amp's eq that way. Modern bass amps at their most common settings are not designed to pump out that 60's tone we're trying to get. If possible go straight for a Fender tube amp simulation, or run it through a guitar amp to try it out.
Do NOT run bass through a vintage (or even a modern) tube guitar amp!
While some may handle it, I learned the very hard and expensive way that vintage tube guitar amps like the one I tried it with are not set up to handle a thumping bass. It sounded great for a few minutes, then needed a few hundred worth of repair to fix what was blown out. The exception would be an old piggyback Bassman, or a vintage amp which has been modded to be able to handle bass versus guitar. That's just my take from experience, I'm sure you'll find many players who might disagree.
Fender style bass, tone and volume at 10 - palm mute - heavy pick - Fender tube-style guitar amp sound (or guitar amp in general) - reverb - slapback tape delay.
Give it a try and report back on the results! I'll adjust anything that doesn't work...