Sure it's not the solenoid on the starter motor. The one on my WB would start it from cold every time. But when you went to start it hot it would do the old click click and nothing. A good tap with the wheel brace on the solenoid and away it went. Needed a rebuilt starter motor.
Also when you say the battery is showing good charge, is that showing 12 volts? The battery could still be no good.