For safety, consider running the fuel pump later from a tap on the oil pressure sending unit. If a fuel line gets cut in a crash, the engine will still run for a minute until the float bowl runs empty in the carburetor, but at least if you're sitting in a courtroom, that will show you understood the potential danger from a fire hazard and made an attempt to address it.
The electrical problem sounds like bad cable connections at the battery. 12.3 volts is too low. 12.2 volts is what a good but discharged battery will read. A good, fully-charged battery will read 12.6 volts. You might want to charge the battery for an hour or two at a slow rate, but I think there's more than that going on.
Turn on the head lights so current is trying to flow, then measure the battery's voltage again right on the top posts if it has them. If you only have the side terminals, measure between the engine block and the larger terminal on the starter. If you still find 12.6, (or 12.3) volts, move the negative probe to a paint-free point on the body. An unpainted bolt head is usually a good place. You should find the same voltage there. Based on what you find, we'll figure out where to go next.
Tuesday, September 1st, 2015 AT 1:44 PM