1968 Barracuda - Where is this bulkhead disconnect?

I don't know what might have went wrong with your reverse lamp socket repair. It looks straight forward enough.
I can tell you that both inst' panel circuit boards are chassis grounded through their mounting screws into the same single inst' housing and dash. So if one side has ground and works the other side does too. All 4 lamps burned out at the same time isn't likely but... who knows. To have broken contact pins at this circuit board is more common but even then, all 3 contact pins broken is rare. I'm going to guess that connector is off, left off in a previous service.
Even if you removed the drivers seat and the headlight switch, and removed the one screw holding the fuse box in place, and set the fuse box and that wiring out of your way so you could get a hand up in there... the connector doesn't have a key or guide of any sort. I wont say can't 'cause cant never could. I haven't tried to access this connector any other way than pulling the inst' panel out of the dash and tilting it face down. AAAAand that "tilt" goes back to the broken pins. This connector and the upper pin are too close to the top of the panel. Tilting is how bent/broken happens. The service manual says to pull panel outward and tilt it. It doesn't say how far outward before tilting.