Your concern:
My problem - I will be growing a lot more plants than the waste water from the fish will support.
may not be a valid one. There's a group called Nelson Pade not far from my house that raises an incredible amount of veggies on a small number of fish. The infrastructure is expensive but there's no physical reason you can't put an entire acre in aquaponic grow beds.
There are some fish (tilapia, carp, some catfish) that can live entirely off dried and pelleted manure, though they won't grow as fast as with fish
feed. If you grow your other animals organically it's not a problem in the water. You can recirculate it just fine.
You can also make a duckweed
pond and pour your manure into that, then harvest the duckweed and feed that to the fish.
Or you can feed them dog food.
Not sure if you're wanting to do prepping or not. If not (really either way) the most effective way to use aquaponics is to grow a highly nutritious and valuable crop to trade for something with more calories. Or you could do partial water changes every so often and use the nutrient-rich water to water sorghum.