And another thing you could try is running whatever you decide on through TH2, GR5 or another guitar sim to get more crispness/gain. That could draw the attack out as well and give you more options to get a P-bass + gainy bass head sound.
I actually kind of like GR5 for bass a little better than TH2. Which reminds me... there is also the trick of triple cloning your bass track and having a clean sound, a sub bass sound and a fuzz sound then blending them all together. Works great.
With this method you could use the clean or probably more preferably the fuzz sound to EQ in more of the attack which could mimic the clicking.
To take that even further you could use the VS-64 Multiband Compressor (or whatever it is called) to block out all frequencies except the attack so that way you have a track dedicated just to the clicks. In theory (because I am kind of pulling this one out of my arse) the compressor itself can get the attack sounding a little sharper.
Basically you need a realistic clean electric bass sound with a bit of click on the attack. From there you hone in on that attack and manipulate with whatever tools you have to get it to stand out the way you want. Tons of options for this.
aaaaand one more thing to toss into the mix is the Saturation Knob. If you set it to Keep Low (which leaves the low end untouched but adds saturation to the high end) it should make the click more pronounced as well.
These are all just things to try. Some of it may not work at all but if I was attempting something like this these are the things I would play with. I'm about to fire up the DAW so I'll check on the Alembic path for you. Cheers.