Using just plants and water is almost useless in terms of cycling. The plant surfaces probably have traces of the bacteria that you need but they are not nearly enough to safely put fish into a tank. If you have a heavily planted and thriving tank, like this one, you don't even need to cycle it. The plants will absorb all nitrogen from your fish. That is one of the principles involved in Diana Walstad's NPT being populated on day 1.
Note: this tank is lightly planted when it comes to relying on it to remove nitrogen. It needs to be cycled to really work.
Why not simply split the media between the existing and new tank filters, or you could keep the new filter wet while you move it so that it stays cycled. You can steal about 1/3 of the media from a mature filter nad usuaslly have no problem in the old tank.v That will give you a nice start to cycling the new tank and may permit you to stock with a few fish.