Alright, sorry for calling you out since you werent trying to flame F1, :grouphug:
Adding more media is rarely a bad idea, especially with your flow rate so high you wont suffer much, so I say go for it. And whats your pH? If your ammonia is shifting your pH should be moving too if it shifts enough, unless there is something in the water that is giving a false reading, but I dont suspect as much.
My guess (and its just a guess) is that something happened the water (foreign material, lack of oxygen, circulation, etc) that killed off the ammonia breaking down bacteria, but the denitrifying bacteria survived because they werent susceptible to the event. They may not have been starved for long enough to die off so your ammonia isnt being converted, so it spikes, the nitrite isnt being created so it levels. Nitrite would consumed, but there isnt any, so no new nitrate is being created.
At any rate, i think your idea of adding media is good. I also think youve done enough water changes and general maintenance that you might be ready to just let the tank cycle. In that case keep an eye on your ammonia lvl. Keep us posted on your progress. Ive never heard of anything wiping only one set of bacteria before but for all I know it may be something people have encountered before and that may be what happened here.
If I can figure anything helpful out I'll post it when I can. Good luck and let us know what you find