Your best bet would be to apply for his pension records. The fact that he applied and was turned down indicates a file was opened. The pension file would have the documents indicating his stays in a military hospital, the diagnosis, the claims of illness made by the family, and the government's reason for claiming these not related to his service. We explain how to apply for these at http://www.spanamwar.com/geneaol2.htmhttp://www.spanamwar.com/geneaol2.htm
Some tropical diseases were such that they hit the men hard after they were out of the area of combat - After they were discharged and back home. The government debated these cases. There was a high rate of death among men returning from the jungle battlefields in the years immediately following the wars (Spanish American and Philippine American) that are not calculated in the known war death toll.
I also have a relative who died following his service. His family was eventually granted a pension, after first being denied for similar reasons.