I reckon chickens atre generally better than fish because of all the extras they give. However aquaponics has its advantages.
I use a base of chicken manure, nettle tea various vinegar extracts and fish amino acids as my fertilizer. Then once a month I add a little generic soluble fertilizer to balance the micronutrients.
The fish haven't suffered yet. They're goldfish so they're tough. It's one reason we're changing styles because we don't eat the fish and that's a big part of aquaponics yields.
That was my first consideration with regards aquaponics, but I feel like producing the fish for consumption would need to be on a fairly big scale, so probably wouldn't be worth it for the size we could do.
In my experience, chooks are more robust and you can let them run amok. Protein per kilo is higher with fish but they're more fiddly to manage and nowhere near as much fun!