No need to transfer the water as well, there's nothing but nitrates and other wastes in it. Just move the old filter and if possible the substrate as well, then refill and add the fish back in once the water is at the right temp.
When I switch/move tanks I will use a 5gal for all the fish, often times a good portion of that water goes back into the tank.
(i like transferring about 30-50% of the old water just to avoid any potential shock to the fish so it's more like a large water change... but it's not necessary. 100% change is fine as long as you follow the advice above )
good plan on transferring the current filet with all media. that's what's gonna make your tank stay cycled & safe. if you do see any blip in ammonia it will be brief and probably easily managed with prime