Here's another option: Park at Mammoth Lakes, and take the ESTA (Eastern Sierra Transit Authority) shuttle from there to Lone Pine. It runs that route M,W,F leaving Mammoth Lakes (McDonalds) at 7:35 AM, to Lone Pine at 9:40 AM. (ESTA 395 Schedule)
(scroll it left and right to find the stops.)
You can park along the road just east of the Mammoth Mountain Inn while you hike, and the car would be available mid-hike when you come out to resupply from Reds Meadow.
At the end of the hike you could then take the Yosemite YARTS transit from Yosemite Valley back to Mammoth Mtn Inn. It runs 7-days, leaving YV Visitor Center at 5:00 PM, to MMI @ 8:51 PM. YARTS Hwy 120/395 Schedule