Follow Beasore Road until the turnoff to Cold Springs Camp Site/Rest stop near Chilkoot Lake. Follow a dirt track as far as your car can take you, but know that the final section to the shore of the lake requires 4X4 and high clearance.
There are unlimited ways to approach this peak in the late summer and fall, with access limited due to high water/snow levels in streams during the winter, spring and early summer. There is no formal trail and almost every approach involves quite a lot of bushwhacking. Do your best to stay to the South of the lake on the approach, as the brush and chapparal are a bit less dense on that side. Once you reach the base of the peak, the bushwhacking evolves into rock hopping with a few sections of friction climbing up 30º granite slabs. The final section to the summit plateau and the summit block involve easy class 3 scrambling. Marking the true summit you will find a USGS marker and a summit log filled with the names of past ascensionists. Sign the log and return it to the jar for safekeeping!
The full route is roughly 5.5 miles and 1,000 ft of ascent but will feel like much more due to the lack of trail and slow progress.
No red tape.
Late summer and early fall are the best time of year to climb dry rock in the Sierra, but it is possible to climb Little Shuteye Peak in the spring as well. Be ready for higher water levels and some snow if you attempt in the early season.
There did appear to be campsites near Chilkoot Lake in the Cold Springs Interpretive Campsite, but consult the backcountry camping rules for the Sierra National Forest.