Basically any pump can be run off of battery with the right setup. An air pump will give you more water movement per watt so I'd consider that. Remember you're just trying to move water, it doesn't have to be raised up into fountain. Gas exchange happens at the surface. Raising water above the water line cost more watts.

For bio filter to handle ammonia for your size pond I think the pondguru has some great ideas like this one. He has others. Moving bed is a great bio filter so it can be small and do a very good job.

Solar is a problem because generally O2 is a bigger problem at night.

Fish don't get burns from ammonia, they die long before burns could develop. That's not to say ammonia isn't a problem. Ammonia is easy to test for. Obviously 20 gal is small and so water changes would be expected to be an on going task, although I don't know the number of fish.