Drop the new sponge into the cycled tank and let it run for a couple of weeks. Bacteria will colonize it. The thing to realize is that the total bacteria in the established tank will not change much, it will just end up spreading to the new sponge as it will be a more hospitable home for the bacteria than the existing filter. This is because all media naturally clogs over time which means parts of it become less habitable. One the other hand the flow through the new sponge creates a very nice place for bacteria to live.
What happens as some bacteria die off in the mature filter, a few bacteria will find their way to the new sponge and finding it to be a nice home will use the available food in the tank to create a new group. After a couple of weeks you should have sufficient bacteria built up in the new sponge than you can move it over to help jump start the cycle in your new tank.
You can also move substrate or other decor over and also get a jump start that way. Contrary to what one reads on many aquarium sites, a bunch of one's bacteria will be found outside of one's filter(s) in most tanks.
The actual bacteria end up living in a bio-film and not in the water. The action of current combined with how the film forms allows bits of it to be sheared off and these bits form the basis for the bacteria colonizing other areas of the tank. This is also how the bacteria initially get into one's tanks when no seeding is done. The bio-film containing the bacteria are all over the water supply system and household plumbing.
Many of my tanks use sponges in some fashion, especially those for breeding and grow out. What I discovered is when there is a power outage or power blink, power filters may not restart for various reasons, but the air pump or blower will and that means the sponge filters will as well.
There is no need to move over the water in terms of the nitrifying bacteria. Solid surfaces are where they live.