The first thing to understand is that adding the filter will not create more bacteria. it will spread out what you have. Because you are doing this in a stocked tank, there is nothing you can do to accelerate the process. However, if you have the second tank, you can do things differently which would let you speed the process.
By taking some media and/or gravel from your cycled tank and putting into the new one, you can then proceed with a regular fishless cycle dosing ammonia. Because you are seeding to start from the existing tank. The new tank will cycle faster than it would otherwise. Plus you do not have to wait for the two weeks or more it would take to get a decent amount of the bacteria living in the new filter before moving it over.
In the end, the effect on the established tank will be pretty much the same- you will remove about the same amount of bacteria whichever method you use. The major difference will be in the new tank. It will cycle faster and will be able to handle a full load of fish when its ready. Merely moving over the new filter with some bacteria now on it would mean either finishing up with a fishless cycle anyhow or else stocking almost as if you were doing a fish in one- adding stock gradually.