I have an in-law with a father served in WWI was in boot camp about same time and never left the states. The war was over Nov 11, 1918 with limited troops left till about 1920.
There is 3 book set complied in 1920 titled "Soldiers of the Great War" by W.M. Haulsee.
It lists soldiers that suffered causalities by state of residence and type of casualty...killed in action, died of disease, died of wounds, died of accident, or wounded in action. Also gives rank or position (private, sargent, wagooner etc., and "home town".
This is a free access website...http://archive.org/details/soldiersgreatwa01doylgoog
Ancestry also has it on their database also.
If his reason for being in an institution was from a war injury then he could be listed.
The only "Veteran Benefit" of WWI was a soldiers home "The National Homes for Disabled Volunteer Soldiers" and a Bond payable in 30 years.
The State Archives may have the older records of the institution if closed down. Does the death certificate have a box asking if a Veteran? Is he in the 1930 Federal census with box 30,(asking if a Veteran), filled in with "Yes" or "No"? Are you sure he was an "inmate" or maybe transitioned into a staff member at a later date. If he had problems when he got home after the war, in those days a family member could sign papers to have another family member committed. Maybe he never went back home because of that kind of treatment.