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Modified power management (LiOn instead of NiMH)
I would use two 18650 cells (2s1p), because many Digital Servos can directly use 8.4V (full) down to 6.0V (empty). The BMS needed is very cheap: <>, and you can use a simple DC 9V power supply (maybe with one 1N4001 diode if the output is higher than 9.0V) to charge.

For the Arduino Nano 33 BLE (which I want to use) the needed Step-Down is even cheaper: <> gets you five of them...

I would connect the battery Vcc to an analogue input of the Arduino with a resistor bridge (10k + 22k) to measure the voltage. When it's getting empty and reaching 6.0V eventually switch to some middle gear to avoid having to cycle home in low or high gear. Since you then cannot switch gears anymore, the servo just needs the hold current and not the work current, so the battery should last two more hours till 5.8V when the BMS will switch off power completely.
I'm hoping to upgrade to the Arduino Nano 33 BLE for my automatic shifter. I have chosen the hard option by wanting to add ANT+ communication to a power meter.
The Nano 33 BLE accepts up to 21 volts on its VIN pin. Just be careful when measuring voltages as the absolute max is 3.3V.

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