Since the 7th and 8th are Wednesday and Thursday, you have the best chance of getting a permit to enter on those two days.
You might pack enough food to stay out extra nights, and stay low -- even as low as Lone Pine Lake the first night. (Once you get the permit to enter, you can stay on the trail up to 14 nights - it makes no difference)
If there is still unavoidable ice on the trail when you go, you should either rent crampons -- they are pretty easy to use on the main trail, or make sure you get up and down the switchbacks where the ice would be the worst mid-day, before it freezes solid when shade hits the slope.
