I second Steve's recommendation. There are some very tricky and exposed parts of the trail, especially between Mirror Lake and Trail Camp where people get lost, injured, and killed. If you're unfamiliar with Whitney, AND Snow. You would be far better off, and more likely to summit going later in the year.
Really, those weekday summer walk-ins are pretty easy to get. I have tried walk-ins in excess of 6 times and I've never been skunked. Plus if you keep an eye on the website every couple of days you might be able to reserve the dates you want if someone cancels. That is less likely and much less certain... but still a possibility.
Happy hunting!
