As others have said, you have found the difference between using a modified sine wave inverter vs a pure sine wave inverter. MSW output is a square wave. PSW is just like the 60 Hz power from your home outlets. The difference is obvious using an oscilloscope. A square wave is missing all the odd order harmonics from the complete sine wave. You can't convert a square wave inverter output to a true 60 Hz 120 volt sine wave with any transformer or other circuitry. No battery is going to change the output of your MSW inverter. You need a PSW inverter.
My
200 W PSW inverter easily powers my Carvin AG300, rated at 250 watts when using an additional 3-way cabinet. This would be more than enough for your 60 W AER, and cost me less than $150 (about 1/10th what an AER Compact 60 runs). Unfortunately the 200 W Cotek appears to be discontinued now. There are other brands easily found on
Google, but not having used them I can't make a specific recommendation.