Witgout having heard your bass I would say that a switch only makes hearable what is normal.
The two single PUs are always a little bit louder than these two PUs in parallel mode.
If they are MUCH louder when single, you might have made a mistake and they are out-of-phase? Switch the two J wires (not the P wires). Does it get louder or even more quiet in the middle position?
If this makes it even worse, change the PU hights of one of the two PUs. The J a little bit more up or the P a little bit more down. It can happen that the hight of both PUs was adjusted so unlucky, that the middle position is not so loud (but this is a rare).
To my mind, two seperate volumes (or a balance pot) make no difference to a switch.