Now, since the Amiga lacks software power control, would the whole system just turn off once the battery was discharged to a certain level, or would weird things happen as parts of the board began to lose power?
Laptops handle this with DC to DC converters to produce +5 and +/- 12 volts from some single battery source, usually 12 to 18 volts. It would make sense to monitor the source battery with the computer or your own separate circuit. I once bought a surplus laptop power converter that handled conversion from the battery, from a power supply, and charging the battery.
I believe Kthunder is right. I've never seen power usage breakdown, but, it seems most of the power goes to the floppy drive, hard drive and accessory add-ons. The hard drive can be replaced with a power sipping CF. I think the floppy can be disconnected.
For comparison, older flashlight bulbs drew a half amp minimum, some would draw 4 times as much.
What are you planning to do?