I finished cutting all the barrels! I now have fish tanks and grow beds for three systems!
In the photos, you can see all three systems lined up. I created columns by stacking two levels of cinder blocks on top of one another to create a good height for the grow beds. The grow beds need to be higher than the fish tanks so that the autosiphons, when triggered, will flush the water from the grow beds directly into the fish tank. By flushing the water quickly, the action of the autosiphon creates a strong suction that brings air deep down into the lower layers of the grow beds. I like the idea of the autosiphon for that reason, since the plant roots will receive extra aeration, more than if the water was just drained with a timer. Another reason I like the autosiphons is that there is no need for a timer- the siphon is triggered by the rising water levels. This means there is one less electric component. I believe this will make the system more reliable.
Once the support structures for the grow beds were at the right level, I placed steel beams across the columns to provide support for the heavy grow beds. A final layer of plywood created a more finished effect, that of a shelf for the grow beds. As you can see in the pictures, this height was still not sufficiently high- I would need more distance between the grow bed and the fish tank for the plumbing. I decided that instead of stacking more cinder blocks, which may become more unstable with addition of a third layer, I should partially bury the fish tanks. This would also serve to insulate the fish tanks against temperature swings. The thermal mass of the ground would keep the tanks at a more stable temperature, even if they were only buried halfway. To bury them fully would have created the unnecessary hazard of the possibility of stepping into a fish tank. I decided burying them halfway was a good solution.