Great, they you can absolutely do it. Bacteria only lives on surfaces, not free-floating in the water, but I’d still fill the new tank with about 2/3 old water. There is plenty of surface area on your live rock and it won’t matter that the new equipment is not “live.”
When initially transferring, To be sure to avoid a cycle, I’d use more established live rock from your 400 in the 200 than you actually intend to use long term. After a month or so, take out what you don’t need.
I am very confident you won’t have a cycle going this route. I’ve done similar in the past with upgrades. It will be even easier with a downsize.