UPDATE:
After some unrelated delays I have the final plan worked out. The building inspector verbally approved my plan. I have to write up a description to go along with the before and after floorplans. The details he asked for in the desc that aren't obvious from the floorplans are flooring type, firestop type, air exchanger model, and ceiling height (both the lowest beam and the average joist).
Min ceiling heights, with floor installed, are 6'4" for the lowest beam and 6'8" for the average joist. I should come in just above those numbers but not by a lot.
I will have three building inspections and two electrical inspections. Building inspections will be after raw framing, after the first electrical inspection, and then after the walls are closed. Electrical inspections will be after the wiring is in place but everything is still open and then after the walls are closed. Yeah, that's a lot of inspections for such a small job, but that's how it has to be done.
When all of THAT is done my gameroom will be habitable living space and will count towards the square footage of my house. That will be a good bump of about 30% on total square footage.