I learned about a legal way to approach Dead Indian Peak from Big Creek, a tributary of the Shoshone North Fork. Sean Sheehan provided the information. Sean owns and operates "Wyoming Nature Tours" out of Cody and lives up in the South Fork.
Apparently, there is a legal access across from Green Creek and the red barn that sits up on the hill above the highway about 9 miles west of Buffalo Bill Reservoir.
At a Wyoming Game and Fish sign, turn north down the hill and cross a bridge over the North Fork. A road or trail then connects up BLM and State sections and traverses into Big Creek east of the private ranch that sits in the meadow at its mouth.
If you know any more details about this approach route, please post a reply.