Welcome to LTG! Now there could be a few things...
The harness in the Newark St. Starfire looks like this. So you see the switch has 3 wires plus ground.
EDIT: BigAndyA is talking a Starfire II bass, not the guitar! So the below is not for the bass but for the guitar! My bad!
Our forum owner GAD
shows a drawing of the wiring (here a Starfire III from 2016), so you see where the white, red and green wires go to.
Since you say neck and bridge pup work independently, just not together, the wiring on the switch seems to be fine, so it looks more like a mechanical issue of the switch in the middle position. Means one of the contacts does not touch the other in the middle position anymore. That happens, especially as those are unbranded Korean switches used in the Newark St. versions.
Since the Starfire II is hollow, it is not to difficult to remove the switch and check it. Especially as the switch is not far from the neck pickup hole:
EDIT: That's the guitar not the bass! My bad!
Maybe it is best buying a good quality
"Switchcraft" brand switch from StewMac or some shop like that (the angled version as visible above) and repace the switch alltogether. Mind that the current switch might have a metric thread for a metric switch tip, so when you buy an imperial threaded one you need a
new plastic switch tip as well.
Ralf