My vote is for the BSH. The hotel is waaay better IMHO and in a different league in terms of character and there is waaay more to do in Banff when you aren't hiking. Take the hotel tour or just walk to floors...At night walk down to the South(?) end of the hall on the 7th floor to the Vice Regal wing and see if you get the shivers!

The spa and regular pool facilities are orders of magnitude better at the BSH. The warm outdoor pool is brilliant! Whereas the LL pool facility feels like you are at a Super8.

If you really want to stay at both then 2 nights 1 night is a decent idea but I hate packing and unpacking, checking out/checking in, especially if summer crowds are swarming so I would be more inclined to stay the whole time in Banff and just drive to LL one of the days. You can day trip to Lake Louise and Moraine Lake for lake viewing and any hikes you may wish to do in that area.