You are right that growing massive amounts of algae (Or other plants) takes a lot of fertilizer; however, you shouldn't have to add any fertilizer with such a high concentration of fishes.
One of the main components in high quality fertilizers is liquefied fish shit. It might just be called liquid fish fertilizer, or fish extrusion.
All you should have to do is have water that comes directly from your filter go into another holding tank without a lid. This could be another kitty pool or something like that. It doesn't have to be too deep, but should have a lot of surface area exposed to sunlight. You can experiment with what aquatic plants you put in there if you want something other than algae.
This would be beneficial for two reasons:
1) You would be growing a low cost (essentially free) supplement to your tilapia's diet.
2) The algae or other plants will use the fish extrusion as fertilizer. The plants will siphon off a decent amount of ammonia to help themselves grow. This can act sort of a buffer, or boost to your biological filtration. In addition to using ammonia, the plants will use nitrates. This will lower those levels as well, or create a buffer.
I would not recommend adding fertilizer to the system. I am not so sure that the benefits out-weight the risk in this situation.
