The Native American reservations can be real sleepers in that they have the habitat and may not get as much non resident pressure as in other parts of SD. We haven't hunted Lower Brule but have hunted Ft. Thompson (right across the river to the east) and found roosters, sharptails and prairie chickens there. You must have a TRIBAL license; reservations are "sovereign nations", so to speak, and your SD state license is no good there.
I believe the reservation requires a native guide--At Ft. Thompson they do--and if there could be a weak spot in the hunting it could be in the quality of the guide. We had a good one the first trip, but a bummer in the second (during deer season, which I'd try to avoid if possible).
We stayed in the tribal casino hotel at Ft. Thompson, Lower Brule has one too. Rates were reasonable (as of 2014). Unless you want to drive to Chamberlain or Wessington Springs for meals, you can eat at the casino too. I remember we went to the Lower Brule casino for supper one night and they served a good prime rib.
Just a word of caution: These reservations are close to poverty-stricken , and most have not forgotten Wounded Knee. They're cordial enough, most of the time, but apart from your money, many of them would just as soon you weren't there. Lock your truck and don't go roaming around at night. We have hunted the Rosebud for prairie dogs; same things apply there. Despite all this, we have found the hunting to be good to very good.
As in most situations in life, just use common sense.