To sum up -
Ammonia up - Can you give us an actual number?
No dead fish found so far
No Nitrite
40 to 80 Nitrate
almost no solids in the grow beds and not feeding much
Things you can do for now -
1. Since Ammonia becomes more toxic at a higher pH I would consider adjusting the pH of your top up water to keep the system pH down for the time being. Water changes to lower ammonia coupled with adjusting the pH of the top up water going into the system would be a good way to do this. No more than a 0.4 pH change to the big tank during a single day (although tilapia can probably handle way more without dying it's not a good idea). I would shoot for a final pH for the entire system of around 6.5 to 7.2.
-- Hydrochloric acid (HCl aka muriatic acid) - Add to topup water - adjust this to 6.0 to 6.5 and let stand 24 hours before adding to system. This will lower the system pH gradually over time. Use safety equipment this is potent stuff and it won't take much (I can't tell you how much you'll need to add because that depends on the buffering capacity of your water). Remember - no more than 0.4 pH change to the main system in one day.
2. Put an air stone in the grow bed to see if this helps bring the ammonia levels back down. The breakdown of ammonia to Nitrite is dependent on oxygen. Make certain you've salted to one ppt for Nitrite toxicity in case you get a spike because of this. Note: If your Nitrate levels keep going up you may need some areas with low to no oxygen (something to keep in mind for later on)
1ppt = 1KG per 1000L or 1gm per L
3. Add a 5 gallon bucket Radial Flow Filter to the system for solids removal. This will remove about half of the solids passing through and these will need to be drained once or twice a week (I tend to get sloppy with the timing, what can I say

) Put the solids out on a soil garden or your lawn (mine go to the compost heap or top of the wicking beds. Alternatively you could use a solids digestion scheme to either break them down for more nutrients or going a different route, break them down to achieve lower nitrate levels.
4. Get someone else with a test kit to test verify your ammonia readings.
Did you or do you still have any algae in your water? Algae and some other plants can use ammonia or nitrite without waiting for the conversion to nitrate. If you had them and lost them, I suppose it's possible there would be a lag for the bacteria to build up and compensate for that loss.
Regarding the Tilapia numbers - they seem more durable than most fish and you can probably stock higher levels because of this but I think it would be a mistake to compare to cookie's numbers. Mr D. is correct that these should be related to the filtration volume you have. In Deep Water Culture root surface area plays a role in increasing filtration capacity - I'm pretty sure this has an effect in media beds as well. If you know anything about tomato plants and you look at what cookie is growing and how much space they are growing in you'll see why I mention this. Then too, cookie is in the tropics and tomatoes love warm humid nights. Not meaning this as a knock to either of these guys, I think they're both pretty sharp.
Hope this helps