I've not seen a report or document that answers your queston as a first hand primary source document. I am convinced he was in his cousin JW Cypert's unit, part of JT Coffee's Regiment with Price in the 1864 raid. Every indication that family geneology researcher Lyndel Storey (Batesville, Arkansas) comment that he was killed in KS October 1864 fits. If you note, the ajudication for James B Anderson's will indicate Harrison Passion Anderson was his son and was dead at the time of ajudication of the will as it relates to the "heirs" of Passion Anderson. The Cypert-Anderson family connection is strong and proven.
James P Anderson that served in the EMM that Elmer refers to survivied the war and received a pension from Missouri as an invalid in 1899.
NPS entry as follows for possible match
H.H. Anderson (First_Last)
Regiment Name 6 Missouri Cavalry
Soldier's Rank_In Private
Soldier's Rank_Out Private
Film Number M380 roll 1
I've seen Cypert spelled Scypart and Scypert as well.
There is a MO SOS entry for a Harrison Anderson that was captured at Lexington that took the Oath and moved to Kansas per the History of Lafeyette Co. but nothing more is known or stated and no contemporaneous records confirm the story.