If the key objective is climbing Whitney, then Kearsage Pass to Whitney Portal is great (we did it last summer: viewtopic.php?f=1&t=8265), but doesn't have too many options for doing a substantial portion off trail, other than day hikes to nearby peaks or basins like Wright/Wales Lakes. The first day to Kearsarge Lakes isn't too bad if you acclimatize by camping at altitude the night before.
Kearsarge to Whitney seems like a good fit for older, experienced scouts. There are in fact some good off trail options that would make for a memorable adventure. After Forester Pass, the kids could do a loop into Milestone Basin, in the upper reaches of the Kern watershed. There are trails here, but plenty of options for off trail hiking and peak bagging as well.
Another option is to hike into Center Basin, over Junction Pass, and then over Shepherd Pass. This avoids beautiful Forester Pass, but sets the kids up for some memorable peak opportunities -- Mt. Tyndall, Junction Peak, Diamond Mesa, Mt. Williamson. Most of this route is trailed, but it is lightly trafficked compared to the JMT.