Faulty battery, bad alternator, or both (photos)

NO. "bypass" as in run 12 volts directly to the choke assist is not the plan. Don't bypass the controller. Disconnect the wires and leave them off. The original thermal activated choke is still in the well. The electric assist can stay in there too, just don't supply power to it.
Do this... After you remove that old school controller, bend its tangs back and open it. You'll see.