Get two multimeters with logging option
Get a very low value resistor, like 0.01 ohm .... put it in series with the battery ...
Measure the voltage across the resistor AND the voltage after the resistor and you get the current and voltage... log them. V = IxR ... you have the R, you have the measured V, you have the instant current value.
Alternatively, you can get one of those programmable power supplies which can also report voltage and current to PC or some logging device (through serial, usb , network).
You will have a pretty much constant voltage, unlike a battery that will lower voltage as it discharges, but you can calculate the power consumption over a long period of time and determine the average power used over that time.
You can also fake a battery by lowering the DC output every 30 minutes or every hour by some amount, which would account for the efficiency of your dc-dc converters (if your circuit has any)