DC amps fluctuating when measuring?
A load needs to be tested under known conditions, minimizing as many variables as possible. When trying to determine maximum power requirements, those conditions need to simulate a worse-case situation.

In your case, you would set up your experiment to operate with an input of known frequency and amplitude(the "Control"), and a possibly varying gain(the volume level; the "Variable"), to see how much current the circuit requires at maximum or near-maximum volume levels. When measuring current, the meter should be connected in series between the power source and the load.

If you don't have a known-stable input source(for example, a function generator or audio test-tone), then your experiment cannot produce usable results.

If both the amplifier and Bluetooth receiver are powered by the same low-voltage input, then there is certainly a regulator in place before the Bluetooth/controller circuit, and amplifier circuits are generally very lenient in their operating voltage, so you should be able to get away with a 6 to 9 Volt supply at 1 Amp of current if the amplifier only puts out 3 Watts or less.