I do specialize in electronic design. Most of my designs are in analog electronics. About 50% of the time, I'm fixing an instrument or an electronic system in an enterprise. I did lots of modifications to existing systems to add functions and so on.
The simplest solution would be to put 2 li-ion 1 A/h in series and try your device at that lower voltage (8 volts fully charged and 7.2 volts nominal). As far as I can tell, this will work with almost all devices that need 9 V.
You could use led headlights to save in energy consumption. Yes, you can use an extra-battery of the same type in parallel. Although when the motor is on, the alternator is charging the battery, that battery is still connected and will furnish current if need be. Your alternator will take more...