My only warning is to avoid filling up with gas north of Lone Pine. It is expensive to truck in the fuel, and Bishop seems to have record breaking high fuel prices!
It is much less expensive to fill up in Bakersfield, and top off in Ridgecrest area. The fuel pipelines do not go north of Ridgecrest, it must be trucked in from there.
You should find that highway 58 is open to traffic, and normally free of snow, except for a snowstorm that might delay your passage for a day or so.
I am not sure about highway 178. I have been through that area the last week of December, and that year there was no snow, it was comfortable to be in shorts while out of the car in the daytime. Yet my friends have reported snow in nearby Jawbone Canyon!
I did drive up to Lone Pine one December, and found that the road to Mt Whitney Portal closed due to snow. The campground at the base of the mountain was open.
There is a campground called Horseshoe and it is 10,700' elevation! That was closed for sure!
Many areas around Lone Pine where used as movie sets by the Lone Ranger, and Gene Autrey, and others.
Have fun camping!
Fred.