Peltiers are inefficient in the sense that you will consume way more power per BTU than say, an air conditioner. That said, they do have their place, as they are solid state and probably very reliable compared to a standard refrigeration loop.
So depending how cold you need to go, it might make sense to get a small fridge or air conditioner, and circulate the water through it. You could make a heat exchanger. Depending how smart you want the system to be you could have various temperature sensors that control the pump and the chiller (whether it's a fridge, freezer or maybe even AC unit).
I'm guessing it does not need to be super cold, so I'm thinking a bar fridge would work. Setup a water tank inside the fridge, then coil up some copper tubing inside the tank to act as heat exchanger and have the fish tank water circulate through it. The idea is the tank inside the fridge acts as thermal storage. Then have a temp sensor in the fish tank that activates the pump when the water gets too warm. I imagine not shocking the fish is also important, so by using this indirect cooling method it should work decently while having good capacity. Suppose you could have the fish tank water go directly inside and out of the tank that is inside the fridge, but I feel it's probably best to keep it separate.
Disclaimer: I never did this before, so I can't guarantee it will work, but I'm thinking it probably would, worth a shot unless someone can confirm it won't work.
You certainly nailed one very valid idea Red Squirrel! Thanks!