All worthy mountains!
Split Mtn is a off trail after Red Lake, and the road in to the trail head requires a high-clearance vehicle. (not 4wd, but almost). Certainly one you could do, but... not top of the list.
White Mtn is a definite must-see mountain. Not nearly as difficult as Whitney. There is a 4wd road to the summit, but you have to hike it. The road in is 15+ miles of dirt with washboard in places, so it's a slow drive.
You might consider Mt Langley, out of Horseshoe meadows, take either New Army Pass or the older un-maintained Army Pass. Once on the pass, there is no trail, but it is just a walk (sometimes through loose decomposed granite) to the top. Quite a bit longer than White Mtn, but not as far as Mt. Whitney.