bimmer -
I'd try a few different things.
First, I believe your pickups should plug into the EQ board using molex connectors. If so, the first thing you should try is swapping the pickup leads. This will cause the bridge pickup to go through the neck pickup's EQ preamp and vice versa. After you do this, if the beeping is still on the same extreme of the blend pot, then this means it's the preamps rather than the pickups.
If the problem is in the EQ circuits, then I would definitely reduce the trimpots (actually, I would do this in any case). Your Alembic can generate an enormous amount of output level compared to most basses, so your amp doesn't need (and probably doesn't want) such a high input level. More on this below.
If the beeping switches to the other extreme of the blend pot than before, then the problem is in the bridge pickup - perhaps it's gone microphonic in your playing conditions. I think this is less likely than a preamp/EQ problem, but it's possible.
When you crank the trimpots all the way up you radically increase the output gain (we're talking about more than +20db from typical levels I would guess which is huge). This will also have the effect of reducing the dynamic headroom on that preamp channel, so you'll probably drive the preamp into clipping before it gets to the output jack. You might prefer the edgier/distorted sound, but there are probably better ways to get it after the onboard preamps. A clipped signal has the effect of creating more high harmonics, which is more content for the (stressed) preamp to amplify, which makes the problem even worse.
I suspect that this is actually the core problem that you have, since the problem diminishes when you filter off the high freqs. A lot of times that you get these beeping and scratching noises are when you are overdriving the preamps with frequencies that you can't hear and that your amp can't amplify. The Alembic circuits are a purist design which doesn't seem to filter off lows below the fundamentals or highs above the regular harmonics. So, I woudln't be surprised if the problem is that your setup is creating a lot of 10KHz+ content at high boost and causing the preamp to overload for feedback creating the beep.
If the characteristics of the beep change when you change the battery, this is an overload problem. Changing the battery will increase the headroom of the preamp, raising the level that it can operate at before it clips and causes this feedback problem.
This may also be an overload problem past your bass, in your amp or effects if you use any. You can easily test this by plugging into another amp (even your guitarists' or keyboard amps). They'll all have slightly different input sensitivity which will cause the frequency and pitch of the beeps to change. Just as the onboard preamps can be driven to overload with high output levels, the preamp of your bass amp can also suffer exactly this problem.
If you like a more grindy, distorted tone, you'll probably have better luck returning the trimpots to their mid-range, then finding a post-bass effects unit that gets you a workable tone. I know this might not seem very satisfying compared to what you you're doing now (I generally don't care much for external effects units). But it's better than beeping! Actually, if the problem is supersonic overdrive (overdrive happening a range that's higher than you hear), you may be able to fix the problem with a low-pass filter set at 5KHz or so.
If, after trying all this stuff, the neck pickup preamp is still beeping, I think you may have a failed op-amp. If the IC is failing, it will exhibit the same sort of problems we're talking about here - reduced headroom, feedback and resonance, etc. This is a relatively inexpensive socketed component in most Alembic boards, so it's not too horrible to fix either.
There's actually no filter caps in your preamp boards. Filter caps are part of the power supply circuit that converts AC to DC power. Your preamp is powered by a battery which is DC. There are caps that are part of the EQ circuit network, but if they fail the tone control would probably become completely non-functional. It would be hard to test the caps without removing them from the board (you would do this with a big, expensive filter cap in your power amp, but not with the little ones in the EQ).
More food for thought, anyway,
David Fung