Is the tank established? It's a myth that beneficial bacteria only lives in the filter as in the established tank it's on/in every hard surface, especially the substrate.
When moving to a new filter in an established tank, you don't really need to have both running, but you might setup the new filter, then rinse or suspend the old media in the tank for a week or so to be conservative.
In my established tanks, I aggressively clean filters as they are primarily just mechanical filters. Food for thought: The Very Best Aquarium Filter