You have already answered many of your questions.
An op amp on is own would not be enough...but and op amp
and buffer would be no problem.
An op amp and buffer will not fit behind your ear.
Lets assume your rig draws 500mW (probably conservative).
To run for 24 hours, you would need a 12 AH battery.
That will not fit behind your other ear.
Unless you use some kind of boost converter to increase
your voltage, you battery will need to provide the peak to...