I don't know much about bonsai, but aquaponics, like hydroponics, is about super-enriching nutrients for plants for fast, full growth. Bonsai purposely tries to keep trees and bushes from growing to full, normal size by different means, also called dwarfing, so I sort of feel that both systems are at cross-purposes. Then again you might develop something totally unique and beautiful.
I expect setting a bonsai pot into a grow bed in such a manner that the pot floods and drain regularly would be a wonderful way to keep a bonsai happy. The one bonsai instruction book I read said that the best way to water a bonsai was to submerge the pot in a container of water several times a day. At the time that seemed just way too much trouble but now that I have an Aquaponics system, your question brings to mind that it would be a fab way to care for the bonsai.
Now it would still require lifting the pot out and root pruning regularly as well as keeping the top pruned but I think it could work with the right types of trees.
I think it would be worth a try if you are already trying to do some bonsai anyway.