CanadaDH

The BSH has a ton of different room configurations. Some are renovated, some are old. Some are large, some are small. Some overlook the golf course and rivers, others look up at various mountains, and some probably overlook the driveway. The hotel is huge. If it is the view that you're primarily after, you will want the main building, not the gatehouse, because it's higher up (in spite of the fact that main building rooms have lower floor numbers than the gatehouse floor numbers). Therefore, whatever type of room you want (Fairmont, Deluxe, suite, etc.), go for the "mountain view" descriptor and not "Gatehouse".

Is it worth $200? Only you can answer that, as "worth" is a subjective and personal thing. When I visit the BSH, I request a specific suite right down to the room number. Others are happy wherever in the building they are placed.

Edit to add... you specifically mentioned "Studio Room". If that's the room you're considering, it's more than just a view upgrade over the base room. That particular room, which is a higher category than a Deluxe Room, is more like a 'small junior suite' (but not quite as large as the various suite category rooms). You'd be getting a slightly larger room than the base Fairmont Room, in addition to a better view.