Converting Mono Output bass to Stereo Output

It’s a P bass clone?

The typical stereo wiring provides separate outputs for each pickup, which obviously requires more than one pickup.....

 

I'm very confused. What is your objective?
P bass, single pickup... do you want each half of the pickup having a separate output? Or something different?
Or do you want two outputs so that you can send each to a different amplifier or chain of FX perhaps?
It's not clear to me what you're trying to do.

 

ouch, be careful with those irons! At least it wasn't your fingers you melted (been there, not fun at all).

I'm still not clear what it was that you wanted: did you just want to separate that one signal from your bass into two identical signals? 'stereo' here just means two channels, sort of, the question is what do you want to put through each. If all you want is split the signal in two, you can do that easily, but unless you intend to treat each channel separately (different FX, or amplifiers or...) I am not sure I see the use (and you can always use an external little splitter/mixer).
If you want to have different outputs on each channel you need to decide which. You can route one half of the pickup to a separate channel (not sure I'd want that but that's just me). Or you can possibly have the direct signal through one channel while the other carries the signal after it's gone through the vol/tone control. Or you can add other circuitry to one of the channels. Or...
Just saying mono -> stereo doesn't mean a lot, especially when talking of a single pickup bass, you need to tell us what you mean, and maybe can come up with something interesting by the time you get a different jack.
Maybe?

 

This seems like a case of someone finding a nail and looking for a piece of wood to drive it into.