Overnight on the (non-trail) Mountaineers Route (MR) of Mt Whitney is outside the lottery. (
Day hikes on that route require the same day hike permit as the Main Trail, so IS in the lottery.)
The Overnight MR permit can be reserved starting 6 months before the date of entry (day you start your hike), like all other trails in Inyo National Forest's trails, via Recreation.gov. It is identified as "
North Fork of Lone Pine Creek JM34" Six permits are reservable for each day, and 4 more are held back for walk-in requests, available the day before date-of-entry at 11 AM in the Visitor Center s of Lone Pine.
On Recreation.gov, the route is found by selecting Inyo National Forest, Permits and Wilderness, Cross Country visiting Mt Whitney, North Fork of Lone Pine Creek JM34.