I've just been reading back over your posts and realised your Betta is currently in a filterless bowl on which you're doing partial daily water changes to control ammonia. In this case, rather than cycling the new filter/tank with ammonia I would move the Betta in and let him cycle it. You'll need to do daily water changes to control ammonia/nitrite while the tank cycles, but you're doing that anyway so overall he'd be better off in the new tank as he'd have more space and water circulation.
Nitrate is the final product of the cycle and is not processed by the filter, so no matter what size filter you have, nitrate will always rise relative to the number of fish, amount fed, etc. It can only be controlled by watr=er changes, but as a lot of the UK has nitrate in the water supply, it will never be lower than the amount you have from the tap. I would do another water change of about 50% to get it down and then increase your regular changes to 25% per week.