For nitrate removal terrestrial plants are the way to go.
I would give greater detail but being the season etc etc.. I will do you a short version and remind me early in the new year, i will come back with alot of detail for you.
Depends how much Ammonia we are talking about at the start, if Ammonia isnt a problem then you can skip over the nitrite question, the role of Anaerobic media and Nitrite collide, so in short if your Ammonia isnt a problem we can go straight to dealing with nitrate and not nitrite.
The most effective method, would depend on roughly how much nitrate you generate in a day. If it was significant then you would go the mist and air injected CO2 into the root ball of a say a tomato plant (could be lettuce dosnt matter for now).
The set up is simple imagine a square section of PVC guttering, this has a fitting to take a pump hose one end and overflow pipe the other. The overflow goes into the tank.
If you produce so much nitrate you need to get rid of alot, then you use a suitable size vivarium mister, the kind that use ultrasonics to create an almost dry water mist, this is blown by a pc type fan into the guttering, the guttering has a lid on top. You can buy these at hydroponic shops.
You blow the ultra fine mist into one end of the pipe, directly underneath you inject very small amounts of CO2, the mist absorbs this as carbonic acid. This simply makes the carbon available easily for the roots.
Now depending on the size of tank, and the amount of Nitrate we are talking about, the details vary but the set up is roughly the same. If the nitrates are not massive then do away with misting and go two chamber guttering and 1/5th flow from filter into the guttering from a tee section.
Without pics its hard to get the setup in your head. So in the new year i will take you through it if you like. That method will outstrip any other type of setup, being terrestrial plant it is also self governing, the plants takes all the nitrate, you grow it SOG method.
So not enough nutrient and the plant dies back a bit, but the roots are fine, its the roots we worry about

. In most situations lettuce plants will do as well but need changing every 8 weeks, toms last 18 months if done right.
Average daylight will do, or if you really have a huge sump filter and produce grams of nitrate a day, then you may need to add light, but rarely. There is alot more to it like buffering just after nitrate removal (easy to automate with a separate section and marble chips).
Messing with Anaerobic chambers is fun, but only needed in pretty special circumstances.