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.
Here's the process I went through to obtain a record for an individual who died at Fulton in 1881:
The person who said many of their records were lost was correct, and the cemetery map / interment records were lost altogether; however, not as many medical records as you may think were lost: Specifically, some records were destroyed between June 1888 and August 1891 (record numbers 4974 through 5479) and Medical Record Volume 2 is missing that contains records from 21 Sep 1853 to 27 Sep 1856 (medical records 0111 through 0280).
In accordance with Missouri Revised Statute 630.140, release of medical records of a deceased person may only be obtained through a Missouri issued court order.
Because of this law, Fulton cannot "pre-check" the records to even confirm or deny whether a patient was ever there, so you'll have to secure the court order first, which is pretty much a matter of sending the correct notarized form called an Affidavit for Information and $71.50 to the Callaway County Circuit Court Clerk. You'll have to call them to see if there are changes or new rules due to the pandemic. [There is nothing the Fulton staff can do about this, there are no shortcuts or workarounds, it is state law and they follow it to the letter].
With court order in hand, you will then need to apply to Fulton State Hospital requesting the records and pay their fees, which, as of August 2019 were: $22.82 search fee (includes first 5 pages of record), $0.59 / page for additional pages, $6.50 for records on a flash drive, and Missouri state sales tax.
This is no guarantee that a record will be located, it's just the process you have to go through to even have them search for one. If it is found, it will be reproduced in the manner in which you choose and sent to you. If the individual was buried on the grounds of the hospital, their medical record number will likely be the only thing that is carved on their stone and the only way of locating them since the interment map was lost in the fire.
Greenville County Sun, 05 Nov 1896, Ben Hughey appointed guardian and curator of Clinton O. Hughey, insane, bond filed on 10 and 14 Sep 1896.
There's a note in the Greenville Missouri Sun 30 Oct 1902, page 4, for the November 1902 term of the Wayne County Missouri probate court that mentions the estate of Clinton Hughey, insane, Ben Hughey, Curator, final settlement.
There's also a 15 Apr 1909 notice in the Wayne County Journal that Robert I. Hughey will sell real estate of Clinton O. Hughey, deceased, to pay debts of the estate.
from the Greenville (MO) Sun 30 Oct 1902, which would indicate that Clinton Hughey died been the census in 1900 and Oct 1902.
Ben Hughey, Clinton's brother was appointed by the court in the mid-1890's to act as guardian for Clinton for the settlement of their father's estate. In 1908 Robert, another of Clinton's brothers, is acting as executor of Clinton's estate. One would have to look at the probate records to first get a death date and second to figure out why the change in executors.
from the Wayne County (MO) Journal 14 May 1908
I visited the cemetery this past Thursday & there's virtually no chance of reading those stones. I'm not even certain most of them were inscribed at all and if they were, the concrete/aggregate they used was soft and has worn away so that there's not much smooth surface left on the stones.
I walked pretty much over the whole cemetery and found only three or four legible stones that had been placed by the families of those buried there, but the great majority of the stones appear to either be unmarked or in such an advanced state of deterioration that they can't be read.
Coupled with the loss of the cemetery documents in a fire in the 1950s or 1960s and it will be virtually impossible to identify a specific grave at Fulton State Hospital cemetery.
The court order through Callaway County is the only route available to see if your ancestor died there.