The reservable Trail Crest exit permits disappear in less than a second on Recreation.gov the morning they become available (15 permits each day, 6 months in advance), and you have to already have your entry permit lined up. It's just about impossible.
When you come in to pick up the Horseshoe entry permits, there might be some exit permits available, but do not count on that. Hikers coming south start a few days before you, so get first crack at those 10 first-come-first-served exit permits.
So plan on the 6-day round trip. I've not hiked in or out that way, but pretty much every water crossing will have camp sites near by if you look for them.
Many people hike in and out that way in 5 days, with the middle day for climbing and descending Whitney's west side. But, if at all possible, talk your group into an extra day -- for high-altitude acclimating. You never know... someone in your group may need the extra time. Do check into using Diamox starting when you arrive in LV.
There are plenty of things to do and spots to see, if you have that extra day (or more!) You're spending enough to get out here, so add as much time as you can possibly get! People here can suggest highlights to see.
