By mid-October, you are getting to a place on the calendar that snow starts to accumulate at high elevations rather than melt within a few days.
I used to go to the Sierra every mid-October and about a 1/3 of the time we had to deal with snow but other times we were walking around in shorts.
With all that said...White Mountain is the better choice because the Sierra will disrupt whatever storm leaving White with less snow or no snow.
It is a roll the dice and take your chances time of year.
