A setup done so that string height follows the radius of the neck can leave the E and G strings too close to flat, non-adjustable polepiece type pickups (or, if the E and G strings are at the desired height, the A and D strings too high).
Accordingly, you could try raising the E string some if the bridge allows individual string height adjustment (loosen the string a bit first before raising a saddle, then retune). Also, even though you say you are familiar with pickup height adjustment, I still want to mention that you could try lowering the E string side of both pickups. This will also effect the A string a bit, but less so than the E. Between raising the E string a bit, lowering the pickups on the E string side (i.e. tilting them) a bit and/or going with a thinner E string (or thicker A, D and G strings), you can hopefully get your string balance issue resolved.
I guess I mentioned the adjustments first because they’re free and quick - you can just get out your screw driver (and/or perhaps hex wrench) and give a few turns here and there and see if it helps. However, a thinner E string involves less setup compromise. While you’re at it, since the strings are the strings that came with the bass, you might as well get a whole new set and start from scratch strings-wise as there could be hidden issues there. Finally, some setup compromise might reduce the E string thickness reduction compromise if needed.
I’d be interested to hear how it goes — I hate string balance problems!
Ultimate quick fix is of course to adjust technique (lighter right hand on the E string). Compressor pedal... Actually, a tube amp being pushed will cover up (or at least reduce) such issues compared to an amp being played with plenty of headroom. A compressor (or heck, distortion pedal) gives that kind of effect, too. Now, I’m not saying the solution is a compressor(!); I was just throwing that out there. Thinner E string and perhaps setup adjustment should get you where you want to go
