Please see the below document from the Missouri State Archives regarding death records in Missouri prior to 1910.
There may be hospital records of the Fulton State Hospital in the custody of the Missouri State Archives.
As you may have already surmised, this is going to be tough.
Missouri did not have state-wide death registration until 1910. Some counties and municipalities did have some records prior to that, but they are scant.
I found Clinton Oliver Hughey b. about 1863 in Missouri in the 1900 US Census of Fulton, Callaway, Missouri where he was listed as a patient in the State Lunatic Asylum No. 1. He does not appear in 1910. Find A Grave Index https://www.findagrave.com/cemetery/2277513 indicates a cemetery at the State Hospital, but I did not find your ancestor. They indicate that a fire in 1950's destroyed the cemetery records, and many graves are unmarked.
The institution still operates as the Fulton State Hospital https://dmh.mo.gov/fulton-state-hospital .
Good luck with your research.