Hello, I think the insulated game room sounds best option. But don't worry about running AC/heat for the battery cabinet, that's not necessary. Extreme cold is worst risk, well the controller and battery bank will generate its own convection heat. It will have a vent that moves air out to relieve that. Either garage or game room should be OK but don't worry about heating or cooling. Think how extreme the temperatures that car batteries go through and last for several years. Having constant room temp would be ideal but don't get too caught up with that.
In extreme climates off-grid people have really large battery banks and build a battery house, like a small play house sized building with the inverters and batteries in it. You would insulate it but that's it. If you put it in a garage you may think of putting in the foam panels in yout garge door and make your weather stripping pretty ship shape but that's all. My cabinet is in a detatched garage and I will heat and cool when I am working in it, but I don't worry about my battery bank getting too hot and cold. The cabinet has good air circulation and evacuation with the dryer style vent.
So, I vote for your option (2) myself. I think that's going to be fine and option 1 would be good too. If you are becoming a dealer I would add the factor of display. If it's in the game room, you can atleast AC it when you are showing it to someone and maybe talking about it without sweating your buns off. So, ability to heat/cool is good there with that option. You can heat/cool when you are working on the cabinet, that's a plus too.