You don't need to boost the nitrites or nitrates. I'm not certain if you fully understand how fishless cycling works. It is a process that can take between 4-6 weeks. You don't add anything to the tank but a source of ammonia. I believe you are supposed to use clear ammonia (you don't want one that contains any surfactants). Most people dose enough ammonia to raise the level to 4-5ppm, as higher ammonia levels can encourage the wrong sort of bacteria to grow.
Bacteria develops that consumes this ammonia and converts it into nitrites. Both of these are toxic to fish and in a fish-in
cycle, should be kept as close to 0ppm as possible. You then get the development of a second group of bacteria that consumes the nitrites and converts them into nitrates. As I said, this process can take a number of weeks.
The tub you are putting your goldfish in is not going to be cycled either, so you might as well keep it in the 10 gallon tank as you are still going to have to be doing regular water changes on the tub to keep the ammonia and nitrite levels low.