In the past, whenever we tried to drive up early and leave early, we would feel awfully wiped out by the time we arrived at the hut. So, I don't recommend that. You really want to be pretty perky for a 10-mile uphill ski tour. One option is to book a bunk ($25?) at the Yosemite Bug, which is a rustic hostel past Mariposa and past Midpines Summit. It is not fancy, but it works. Then, if you leave there around 6:30 a.m., you can count on getting to Badger for the permit and all that. The people who get into trouble on the trail are the ones who start at 10-11 a.m. and then get caught by darkness and icing.
I would certainly not recommend carrying two sleeping bags. Carry the good one and a bivy sack, and then augment it with insulated clothing as much as is necessary. Then some kind of shelter/3-season tent would be nice unless you are good with a shovel for a snow cave. The place to camp is a few hundred yards southeast of the hut. There is a frozen pond. For a snow cave, another couple of hundred yards farther in the direction of the east end of Horse Ridge.
I do not recommend alpine touring skis with plastic boots. Way too heavy. I mean, they would be handy if you are skiing the steep on Horse Ridge. However, a light metal-edge backcountry ski is a lot more versatile, and lighter. Having said that, I will add that the hutmaster skis around up there with plastic boots. However, he "owns" the place, so he can do as he wishes. He has saved a lot of people off the trail when they didn't know their own limitations.