Hey Rob,
Conundrum is right...
Checking it with both your PSUs and cables was good sleuthing. That does point us to the bass.
Excuse the dumb question but, you are tweaking the correct trimpots right? Here's a pic:
It does sound like you have a stereo 1/4 jack. Quick test of that concept would be to plug a pair of headphones into it. If your batteries are good and the preamp board is somewhat cranked, you should hear something and be able to determine if one p/u is left and one is right. It won't be a good sound because the circuit us not meant to drive phones (wrong impedance to begin with). And you may need to exercise the plug a bit in case the jack is corroded from non-use. But this could answer the question about that jack being mono or stereo. Also, if you can hear something, you can get a quick general reference as to the level differences, although it will be very low volume.
Really, the next best test would be to swap the T & B connectors where they attatch to the board... If it were mine I would take my tiny diagonal cutters and see if I could carefully break some of that goop off and free those connectors. But I don't want to send you on a destructive assignment so maybe don't go there.
Do you still have the '78 Series bass? Because if you could free up the p/u connectors on this bass you could plug them into that bass as a test...
Next best thing to try I guess would be to swap the two output op-amps if you have the stomach for messing with tiny pins in tiny sockets. The chips themselves are not very expensive but I don't expect you have any spares laying around. Ha! Be sure to observe their orientation before you pop them both out.
OR ... turn down the output of the bridge p/u to match the level of the neck p/u and run the bass at an overall lower mono output level..........
Jimmy J