What is this most efficient way to power a high powered 6 V and a low powered 12 V device from a battery?
Assuming you have AC adapters for both of these amplifiers why don't you convert the 12V or 6V battery to 220V or whatever voltage AC you use: -



I can't vouch for this being a good design and here is the site it came from.

It looks like you only need to generate 20 watts so maybe also buying one won't be a big expense. To get this circuit and plenty like it google "20W ac inverter design".

Here's another design from instructables