Although I am pretty versed on most of the Sierra, that is one in which I haven't spent a lot of time. I suspect your best source of info on the local area will be from the folks at your inn. That is part of their business, helping clients have a good time. I am sure they will have several good suggestions and may have maps. I am not familiar with that lodge but some will even pack a trail lunch for guests.
I have hiked thousands of miles in the Sierra over the years and never carried bear spray. Alaska, yes but that is a different animal. The Sierra bears are not grizzlies but rather the smaller black bears and do not tend to be very aggressive. Just make some noise as you hike so any bears will hear you coming and avoid approaching one and you will be fine. The only time they might get frisky is if you have smelly food or somehow get between a mother and cub. Don't do those things.
You don't mention when you are going but the weather has been pretty unsettled in the high country lately. It could be raining or it could be sunny and 90 degrees. Best to prepare for both and decide the morning before you go out how to dress.
Have a great time.