Aquaponics is definitely not the best way to grow mushrooms. However, I'm sure you can set up a small "Myco-box" and grow them in that same space. It is extremely inexpensive, requires little maintenance, and can sustain itself forever once started. If you want some more information, I would gladly help you out. I've never grown portabella's but I've grown oyster mushrooms with great success.
Here's a link showing you the entire process once you have an oyster mushroom spawn: http://www.vermontmushrooms.com/indoorcultivationmethods.htm
You can either make your own spawn or order it online. Feel free to message me if you want more details.