You best bet may be to apply for Whitaker's military and pension records. They should give a synopsis of his actual military career. We explain how to do this at http://www.spanamwar.com/geneaol2.htmhttp://www.spanamwar.com/geneaol2.htm
The 4th Kentucky never left the U.S., and existed from July, 1898 to February 12, 1899. It did not go to Cuba, and never crossed paths with the Rough Riders (was not in the same camp, etc.). We have info. on the 4th Kentucky online at http://www.spanamwar.com/4thKentucky.htmhttp://www.spanamwar.com/4thKentucky.htm
Considering his comment about Roosevelt, I checked the roster of the Rough Riders, and the only Whitaker was George C. Whittaker (with two t's), so he does not appear to have served with Roosevelt there. The Rough Riders were in Cuba only for a brief period - June to September, 1898. We have info. on the Rough Riders at http://www.spanamwar.com/rrhist.htmlhttp://www.spanamwar.com/rrhist.html
It would seem his claims were spurious. This is not uncommon. In earlier eras, people could make such claims and it was seldom that they were challenged. That said, did he make any claim to serve under an alias? Was Fess his real name? Does he give any specifics that could be traced??
You are correct that the Spanish American War ended before 1899. It ended on December 10, 1898. The war's fighting actually ended by armistice on August 12, 1898, with fighting in the Santiago area (where the Rough Riders were) ending officially on July 17...just after the 4th Kentucky formed. The troops that served in Cuba in 1899 were occupation forces.
Please note that the obituary would not need to be historically correct. Also, it was written a decade after Roosevelt died...he would not be disputing it, and probably would not have anyhow, even if he was alive.
Roosevelt's papers are in the Theodore Roosevelt Collection at Harvard College Library. You may want to contact the Library and see if any correspondence, etc. exists between Roosevelt and Whitaker.