That sounds like it could be a pickup issue. Ideally, it would be nice to get an ohm reading across each coil individually while it’s acting up but unsoldering the leads could cause it to work again.
I think you could make an in-circuit resistance check, as long as you remember that the meter measures the entire connected circuit and not only the pickup.
In a standard passive Jazz Bass circuit, turn the pickup being tested to full volume and turn the other pickup to zero. Measure resistance between terminals 1 and 2 of the fully-up volume pot, where terminal 1 is the grounded outer lug and terminal 2 is the centre wiper connected to the pickup.
If the selected pickup or its wiring is completely open, it behaves as infinite resistance and contributes no DC path. The reading would then be approximately:
∞ (broken coil) 250kΩ || 250kΩ = 125kΩ
With a functioning 8 kΩ pickup, the expected in-circuit resistance would be approximately:
8kΩ || 250kΩ || 250kΩ = 7.52 kΩ
Rolling the tone control fully down may help the meter reading settle more quickly while the tone capacitor charges, although the final steady DC reading should be essentially independent of the tone setting.
Applying the same general idea to the Roscoe Beck.
Select only the pickup being tested. Set the master volume fully up, and measure resistance between terminals 1 and 2 of the volume pot. First measure in single-coil mode, then switch to series-humbucking mode and check whether the resistance increases.
In series mode, both coil resistances should approximately add:
R series = R coil 1 + R coil 2
Therefore, if single-coil mode measures about 8 kΩ, series mode might measure roughly 16 kΩ (assuming same resistance) before accounting for loading from the rest of the circuit. If series mode becomes open circuit, one coil, a wire, or part of the switching circuit may be open.
You measure this from terminals 1 and 2 of the volume pot.
I am not presenting this as a definitive diagnosis only as a way to compare the switch positions and check whether both coils appear to contribute resistance.