If you go the Cottonwood Lakes route, or maybe Cottonwood Pass, Mt Whitney from the west side is reachable. Two days in, climb from the west, then two days out.
Just beware of the high 10k elevation of Horseshoe Meadows & Cottonwood Lakes. Altitude sickness is a possibility,so you definitely need to recognize the symptoms, and what to do in case.
Out of Mammoth Lakes, there are several trails that lead to good camping locations, too. Garnet Lake to the northwest. Duck Lake to the south.
