I would suggest Shepherd Pass for the following reasons: 1) It is the most direct route into the Upper Kern(about 18 miles); 2) It has very few bear problems if you can make it beyond Anvil Camp the first day and bivy either at the Pothole or at an excellent bivy site about 1/3 mile above the Pothole on the left side of the trail. Look for a small grove of stunted pines. There are several prime sites nestled among the trees. Be warned, this is a long first day, but the views are awesome and you can pretty much count on no bear problems, which is not always the case down at Anvil Camp or Mahogany Flats. Bears are also a problem at trailhead, so keep a clean car if you decide to go this way. This is the only trail on the East side that I know of where bear canisters are not mandatory. The only problems I have with Kearsarge Pass is that the whole area is crawling with bears and you will have to drop down into the Bubbs Creek drainage and then go back up to Forrester Pass (~13,400') before dropping down to Tyndall Creek and intersecting the same trail that goes over Shepherd Pass. It is beautiful, to be sure, but much longer and would chew up quite a bit of your time which, IMHO, would be better spent in the Upper Kern, which is hands down my favorite place on the face of this planet. If you go in over Shepherd Pass, you should be in the Upper Kern by the middle of your second day, no sweat. There is a good campsite on the south side of the lake where the High Sierra trail comes up out of the Kern Canyon, but I would recommend going north to the unnamed lake with 3 bays about a mile north. Several good campsites there and it is ideally situated to be a base camp for beautiful day hikes. Another advantage of going via Shepherd Pass is that the trail goes right by past a side trail to Wright Lakes Basin. It doesn't intersect with the main trail, but if you head for the saddle on Tawny Point, you will pick it up. It deposits you right in the middle of Wright Lakes Basin. One other thing: if you go in via Shepherd Pass, be sure to reserve your permit early as there is a 15 person daily quota and it can fill up early in July/August. Army Pass is the longest option and lots of bears enroute. Good fishing at Crabtree Lakes, though.
Best of luck.