The limiting factor in most recirculating aquaponic systems is oxygen, which solids will consume as they break down. That is why they are usually filtered.
I think the fish waste would make awesome food for the compost extract, but I would make sure that there is at least a bubbler on in the extract at all times, with at least two good stirrings a day.
I would set up a sedimentation tank into which the solids are pumped and allowed to settle, with a barrier and pump or just advantageous placement and a container with a lip designed to pour back into the system at overflow. The solids would settle out on the bottom, the
water flowing back into the system, the slight drop putting a little more oxygen into the returning water, and when the sedimentation tank accumulated
enough, I would add it to a much larger quantity of actively aerated compost extract, in a vortex brewer with a couple of bubblers in the bottom. I would probably use one of those bubble lift systems where you inject air into the bottom of a vertical tube inside the tank which carries the resultant less-dense water/air mixture up and out of the top of the tube, largely because it deals well with solids.
What is the goal of your query, Andrew? Are you looking for ways to improve your compost extract, ways to deal with excess fish waste, or ways to improve your aquaponic setup?
Or is it all, or more?
-CK
A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
-Robert A. Heinlein