So are you looking at basically a big plastic cube filled with water and a plastic tray of plants floating on top of the cube? And this fish live in this cube? Or the fish live in a separate cube below/adjacent to the plant cube?
I have not done aquaponics, but do keep several home aquariums. You are not going to have a healthy environment for the fish without some form of bubbling or agitation to aerate the water in a small, cubicle, indoor system. There just will not be enough individual water molecules encountering new oxygen from the air. Plant roots at the top of the water will not contribute significant oxygen, in fact, plants covering the surface can interfere with oxygen exchange. Submerged plants give off O2 during daylight hours, but not necessarily through the night. If you do not want to connect it to an independent power source, then maybe find a way for the aquaponics system to make use of some excess power from other household sources, or some kind of green power like a small solar cell, or a chinchilla wheel. Or commit to daily manual water change.
Even though your IBC tanks may feel very large, as others have said they are nothing like the scale of a natural pond or stream which would support fish large enough for human consumption.