I am one of those who states an H tank should not be cycled. I have posted this several times on this site with explanations. While many meds will not harm the bacteria, some will and do.
But lets inject a little common sense here. In the wild fish are usually protected by constantly changing water. A perfect example of this would be a high tech planted tank where one must inject CO2 for the plants. But these plants are able to grow fine in the wild without co2 injection. The reason is the co2 removed is replaced by "new" water full of more co2. If the tank were getting pretty regular water change to bring in fresh co2, the injection would not be needed.
Well it is similar for ammonia etc. If you do daily water changes and well as a daily vacuuming, you can keep the water pretty much safe. Moreover, there are a variety of ammonia detoxifiers out there should an emergency actually arise. And ammonia does require some time to show up to where it reads on a kit.
Lets use a it of logic here. the fish went into the tank with "floss". A day later there is 0/0/0. But the fish is certainly creating some ammonia. So if the floss is providing cycling bacteria, where is the nitrate? While that kit may be inaccurate on the absolute level, it does detect nitrate. So we know ammonia is being created, it is not being read nor are the other two compounds. So where is the ammonia? One other consideration is the dechlor being used, does it contain an ammonia detoxifier?
The amount of useful bacteria in used filter floss is minimal. The bacteria need places to live where there is a constant flow of needed nutrients. Good flow through is the key. Floss filled with the gunk it is supposed to trap doesn't not meet this standard. I have been replacing the floss in about 25 filters weekly for 10 years now and never have cycling issues. I am too lazy to clean floss and can afford to spend the money to make it disposable.
If you are that worried about ammonia, then you can throw some stem plants into the H tank to float on the surface. They will help suck up ammonia that might not be handled by daily water changes.
As to big water changes not affecting the bacteria because they live attached to objects defies common sense. It is one thing to change water weekly, it is another to do so daily. The bacteria need ammonia and nitrite to thrive. Keep doing huge daily water changes and they are deprived of these things to a decent extent. Deprived means they do not survive in the same numbers. Moreover, if you are medicating a tank, water changes remove meds which then must be replaced. The more water you change and the more often, the more costly medicating can get.
In most cases fish will be pooping and some food may go uneaten. One needs to vacuum this stuff out daily or even more often, so right there you are doing some amount of water changing. Add to this the fact that a sick fish is much less active and therefore will produce less ammonia than usual. As for the fish perking up some a bit of this may be due to reduced stress. When a fish is ill or injured it tends to hide. It is not able to hold its own against anything. By moving it to an H tank this threat is removed and reduce the strss level of the fish. Stress is a major contributor to weakening the immune systems of fish. Towards that end i have also found that providing the fish with some cover in an H tank is often helpful. I heave a few plastic plants amd rocks I use. Not enough to make it hard to observe but enough to make the fish feel more "comfortable."
The worst outcome in an H tank is we fail to nail down the problem and the fish dies. At this point anything in the H tank must be assumed to harbor whatever killed the fish and needs to be sterilized or thrown out. Plastic plants and rocks can be cleaned, bacteria cannot.
Lastly, sick fish may have trouble with current and also usually need additional oxygen supplied to the tank, especially at elevated temps.. For this reason I prefer air driven filtration in H tanks. Lower flow and helping with O are accomplished this way. I have tons of excess sponges so I use sponge filters and if the fish doesn't make I chuck out the foam. However, one can use an airstone that can be sterilized for reuse.
All of the above is how I do things in terms of an H tank. It is not the only way. While I see little reason for the cycle in an H tank, I see little chance of using cycled media creating a problem. The downside would be the bacteria dies off because of meds and creates waste as it degrades making more ammonia rather than removing it. All that matters in the end is that the H tank be free of harmful things and conducive to recovery.
Keep testing the tank for ammonia daily if you are worried. It is better to be safe. If I am right that the floss has little bacteria, you will see that in the absence of daily water changes and/or some floating plants, as ammonia would start to build up. If I am wrong about the floss, and you are not doing dailies, then you should not see ammonia or nitrite but you should see some sign of nitrate.
In either case monitor the fish for eating and pooping. If it is pooping, make sure the poop looks close to normal. Watch for worms. Sometimes fish get constipated but they also can get bacterial infections that cause secondary problems or have worms or parasites that cause such problems. Also, you do not need a light on the tank and a darker setting should be better for the fish, but you also need to be able to see everything very clearly, so you may want a light that is only on when you are checking for poop and fish symptoms.
Good luck with it.