From a chemistry perspective, I don't think there's going to be a lot you can do about the nitrates without substantially modifying the filter system. However, looking at your tank on Cubic Aquarium's site, it appears that connecting external equipment is possible. What I might recommend is connecting a small cannister filter equipped with carbon and a fair amount of aragonite coral sand. This will act as a "Deep Sand Bed in a Bucket", with substantial dinitrification capabilities.
With that surface area in a cannister, you can then carbon dose the system with vinegar. Given the system's water volume of 20L, very little vinegar would be required (or advisable). I'd start with 0.1 mL per day added with the food. It may take about 4-5 weeks for the cannister filter to get populated with the bacteria that reduce nitrate to nitrogen gas, but I suspect that you will find that you'll have undetectable nitrates in the long term.
You will also need to add some means of removing phosphate. Water changes will do it, but are a pain and somewhat disruptive to the tank's inhabitants unless the new water is very carefully temperature controlled and has the same alkalinity and pH.
A better plan would be the inclusion of a small amount of GFO in a mesh bag in the system's filtration area.
