Your aim is to cycle the filter for the tank where the bacteria are going to colonise and live, not the water.
I stand to be corrected on this but it may be best to allow the filter that will stay in the new tank to develop by itself and be stable as i would imagine the established filter bacteria will otherwise do the work during the cycling process. Possibly take longer to reach the same stage? You will still need to feed the bacteria with ammonia.
You could take a maximum of a third of media or sponge from the existing filter and put it into the new filter to 'seed' it.
I am currently running two internal filters side by side (in an established tank with fish) with the aim of replacing the older one which is smaller. It has been 4-5 weeks and there is no way it would be ready yet.
Unfortunately, nothing is going to happen overnight or immediately.