Nitrifying bacteria grows on surfaces in a bio film.
This takes time to develop. You cannot expect a filter that has only been running a few days to have any major amount of bacteria on the media yet.
The more bacteria you take from a cycled tank the greater chance of that older tank going through a mini cycle. You want to avoid this.
While there is almost no bacteria in the water, here is what you might do to get the most (it won't be much, but will jump-start a new tank)
Do a very deep gravel vac on the old tank. Work really hard to get all possible mulm out of the old tank. Skim the substrate, close to the surface, and deeper.
You will not get so much bacteria that you are going to cause any problems in the new tank. That is OK, though, you really do not need that much.
Aquasoil will keep on producing ammonia.
It does not matter how much or how little bacteria is in the tank. It is the soil doing this.
It takes several weeks to a month for the soil to slow down its production of ammonia.
It does not matter what happens to the ammonia after the soil has released it. The soil keeps on producing more.
The soil is a good source of ammonia for the fishless cycle.
Since the soil will be producing ammonia for several weeks....
And the nitrifying bacteria take several weeks to grow to a good population even if you start with no bacteria....
... then it may help a bit to start with just a little, but there is no benefit to adding more and more bacteria, and risking the livestock in the established tank. It won't make the soil stop producing ammonia any faster.
Play any sort of games you want with the filters, but do not remove more than 25% of the established media from the cycled, mature tank. If you want to swap filters around, go for it. Just make sure that your current livestock population keeps as much of the bacteria as you can, and the bacteria is mostly on the filter media. So keep the media with the fish and shrimp.
A well established aquarium system will have bacteria in the filter, and on all the surfaces in the tank. Since there is a lot more surface area in all the sponges, bio media and other things in the filter, there will be more bacteria in the filter media. Since the next largest surface area is the substrate you would think that would be rich with bacteria, too. But the conditions in the substrate are not so good for the bacteria. So there is a pretty good population there, but nothing like the amount found in the filter.
All the other surfaces put together have nowhere near the amount of bacteria in the filter or substrate.
Some, but not much.
If I had to put numbers on it, I would say
50% of the bacteria live in the filter (including sponge pre-filters).
25% of the bacteria live on the underside of the upper layer of substrate (away from light, but in the best oxygen location, maximum water flow area)
25% of the bacteria live on all the other surfaces put together. Rocks, plant leaves, surface of equipment like tubing, heater surface, ceramic merpeople...