Jim, when I spoke to the handi-ride people in Lone Pine on the phone, they told me there was no Uber driver in Lone Pine. But one might turn up -- you never know.
I think the hassle of getting up to Horseshoe by shuttle or hitch hiking might be the toughest part of your trip. Once you get up there, there are plenty of sites. They are all one-night-only, so they never fill up.
As for July 4, I think snow will be pretty well cleared out by then.