The documentation for Zero2Go indicates that a charger isn't included, eg: "Zero2Go Omini doesn’t charge the connected battery, and you may need external charging circuit like this [links] for rechargeable battery."
What is the best way to do this? Would I pick one of the other input channels, and add a step-down regulator to 5V to power linked charging circuit to charge the battery? If I'm thinking this through correctly, when I connect a power supply to the other selected channel with higher than 5V, the battery would charge AND the system would run off the power supply. Then when power removed from this channel, the battery channel would power the system?
The image below shows an example.
The 5V power supply is for charging the battery. If you also want it to work as a power source of the device, you may connect it to one channel (marked with dashed line). In such case, when 5V is connected, the device is powered by that 5V power supply, and the battery will be charged. However you need to make sure that 5V power supply can deliver the required current.
Alternatively, you may let that 5V supply only for charging, and connect another power supply with higher voltage, to act as the main power supply. When the main power supply is off, the battery will be used.
Of course, you may also use all of them and have the dashed line wiring connected. The priority will be higher voltage > 5V > 3.7V battery.