Are the pickup routes in your bass also abnormally tight so that they are making contact with the pickup shell much of the way around?
I think you are right with that in the general sense of why microphonics can happen, but Bartolini J's are fairly undersized compared to most other brands of J, and they tend to have bigger radiuses on the corners - if a Bart J is tight in a pocket, I can guarantee that most other brands of pickups likely won't even fit in the pocket at all.
However, I also feel like I'm not getting across why I think the problem is in the pickup/screw combination. (Maybe because English is not my main language)
Another example: pickups wired in the bass, bass connected to an amp, holding a pickup in your hand. Lightly tapping the pickup with a fingernail does not make a sound through the amp. Putting a screw through one of the holes makes a loud sound. Leaving the screw and tapping again with a fingernail (the pickup, not the screw!) is clearly audible. Installing the pickup and making sure the pickups and screws are not moving freely: tapping still audible.
Thank you for entertaining us and explaining what you're seeing. I know we're probably not telling you the things you were expecting to hear but hopefully it's helpful.
But, to be honest, that sounds like potentially normal behavior to me, though it's hard to understand the magnitude of problems like this when written in text. We don't know how loud the noises are you're talking about, relatively speaking. Any chance you can get a recording of those different scenarios for us? That could give us an idea of how bad the noise is which might help us understand it.
Any magnetically permeable object placed near a pickup, and then tapped, will make a noise of some kind. If you wire a pickup to an output jack (no bass involved) and then slide a screw into one of the mounting holes, and then tap on the screw - it'll make a sound via the pickup. Or if you tape a screw next to a pickup and then tap it, you get a noise. Even if you put something permeable near a pickup, fix it firmly in place, and then tap the pickup itself (causing the pickup coil to move relative to the other object), you'll get a noise. It's just a matter of how much noise in terms of if it's "normal" or "a problem."
But any time there's a permeable material in the magnetic field that vibrates relative to the coil, you get a signal. That's why things like the "tightness" of the foam under the pickup are important, and why testing the pickup held loose in your hand (versus installed in the bass) is not meaningful. The rigidity of the pickup's mounting impacts how readily it vibrates when it's tapped. A pickup that's really loose, or very tight, or held in your hand away from the bass, can cause enough vibrations to cause an issue. Mounting hardware that's loose, or binding on the pickup, can cause issues. And so on.
The only way it can literally be the pickup's fault is if the internal parts of the pickup move relative to each other when the pickup is tapped. That is very possible, and happens with some pickup designs (sometimes on purpose, to an extent). The actual coil wire in a loosely wound coil can vibrate when tapped, or loose poles or other parts can vibrate. That's why pickups get potted, to lock all the internal parts together as a solid unit. But as mentioned, Bartolini pickups are essentially the most heavily potted design possible, so they're definitely not going to be microphonic in the sense of the pickup itself having a problem (unless you were to break one very badly, but that doesn't seem to be the case here).
