Weather is typically better over in the White Mountains as compared to the Sierra. Checking NOAA weather, the forecast shows the stormy weather moving out by Friday night, with clouds and clearing Sat/Sun. White Mtns are only marginally colder, due to the higher elevation compared to Mosquito Flats.
As for Sierra Permits, you can pick up permits for all the Inyo trails at any of the four Inyo ranger stations. It is only the Whitney Zone permits that MUST be picked up in Lone Pine.