place 2 small water pumps or alike to pump water from one tank to another and back again, ...
This will be a problem. You will never get both pumps exactly equal. In time you will have one tank with much less water, one with much more; very wet floor.
Run the small tank filter in the larger tank for at least 3 weeks, 4 or more is better. Your filter media cycles, the water does not, nor does the tank itself.
This is known as cloning, and how people with many tanks get new tanks up & running. There are some things you need to know before doing this. First, a mature colony of nitrifying bacteria is capable of doubling every 24 hours. Second, capable doesn't mean it will.
Let's say you have a 10 gallon tank with 10 neons in it, and a mature filter. That filter doesn't know if it is running on a 10 gallon tank, or a 100 gallon tank. The media is capable of providing the biological filtration for those 10 fish, irregardless of tank size. You could take half that media, 5 neons, and put the media in a new filter, on a new 10 gallon tank, and can expect to have no problems.
Back to being capable of doubling in 24 hours. You could take 1/3 of the mature media, which will support 1/3 of the fish, and put it in that new tank & filter with 3 newly purchased neons, and expect no cycling problems in the new tank. I always suggest taking no more than 1/3 of the mature media, as capable doesn't mean it will. I also suggest fasting the donor tank for 24 hours before removing media, and feeding lightly for the next week. Less food means less waste produced, giving you reasonable insurance that there will be no cycling problems in the donor tank.
Another reason for only taking 1/3 is that few tanks are set up as easily divisible as far as the bio load is concerned as our example tank. Most often it's a community tank, and the bio load must be guesstimated. Erring on the side of caution, and feeding lightly for a bit gives you the advantage in these cases. I've done this so many times, with so many tanks that I don't bother with any water tests, though I did the first few times. I constantly have spare filters running, never know when I might need to set up a new tank. If I take down a tank due to low stock the filter gets run on a stocked tank, ready to go.
This is also how many people keep filters available for quarantine or hospital tanks.