Jack, I would assume that you are right about changing entry/exit points on arrival. My sense is that the Backcountry Office just wants to know where you are parked and for how long. Since you aren't parking, that should be very doable. I would just call the office and double-check since not all Park Service regulations make sense to me.
The trail from Logan Pass to Granite Park CG is quite easy--7.5 miles of lightly graded trail. If you parked there, I would recommend you hike in to the CG early, hang your packs, and head up the trail to the lookout on the summit of Swiftcurrent Mountain--about 4 miles round trip. It's fairly steep but very worth your time.
Also, if you started at Logan Pass, you could conceivably hike from Fifty Mountain down the Northern Highline and Highline to Logan Pass your last day--it'll be something like 19.5 miles, but mostly a slight downhill grade (with the occassional ups and downs). Then you can catch the shuttle down from there. You can also cut about 3-3.5 miles off of that total and hike down to the Loop from Granite Park where there is a shuttle stop. However, I've never had good luck catching a shuttle there--usually they're too full carrying people down from Logan Pass and no one gets off at the Loop. I've easily waited an hour and half there.
Also, check with the Park Service as to how late their shuttles will be running this summer. Usually they go until the last day of August, and they often offer some service into the beginning of September (maybe through Labor Day?), but you'll want to know that you have a for-sure ride. With you coming out on September 1, that's important to know.