The older gen's like the 4KY units they work on a power in / power out basis, if you put in 12 volts you get out 120 volts.
They have circuitry that 'clips' excess voltage, I.E. if you put 13 volts in you will only get 120 out, not 130, but there is no way to boost low voltage. To protect anything you plug in, and to prevent overload the system merely shuts off power output if it's too low. So if your house batteries are low, say 10 volts, it's enough to crank and light up the motor, but too low to make power safely so it just shuts itself back off as a safety.
First put a voltmeter on the house batteries at rest, then get someone to start the Onan while you continue to monitor the voltage. If levels are good, then you may have a little surface rust on the brushes or just a poor connection.
Often just running it with the button held down for a few minutes, or several start / stops will clear it up.