If you have a good sleeping bag and don't mind a cold night, going all the way to Trail Camp cuts a couple of miles and 2,000' off your summit day. You should be able to find flowing water just below Trail Camp even if the lake is still frozen/snowed over.
The <a href="http://www.wunderground.com/cgi-bin/findweather/getForecast?query=lone+pine%2C+ca">long-term forecast</a> looks pretty promising, so other than being cold up there overnight and into the a.m., weather shouldn't be a problem. (Note that the forecast is for Lone Pine/Bishop, so back off about 4 degrees per 1,000'. With lows down there around 50, you could expect overnight lows down around 20 at Trail Camp, so pretty chilly!)