Trump has every legal authority for Musk’s placement and role.

According to the United States Department of the Interior, a special government employee is "an officer or employee in the Executive Branch of the Federal Government who is appointed to perform important, but limited, services to the Government, with or without compensation, for a period not to exceed 130 days during any period of 365 consecutive days."

https://www.doi.gov/ethics/special-government-employees