Buffalo Soldiers first fought during the Civil War. In case you haven't read a history book recently, the Civil War happened BEFORE World War II.
Not really.
While there were Black units in the northern forces (Blacks in the south, until near the end of the war tended to be integrated with the white troops), the designation "Buffalo Soldier" did not arise until they served in the Indian Wars. The Buffalo Soldiers are more specifically the 9th and 10th Calvary Regiments. (Not organized until 1866).
Those units fought in the Spanish American war under the command of General "Fighting Joe" Wheeler.
The Buffalo Soldier designation can also be applied to two infantry regiments (24th and 25th) also constituted after the War Between the States.
Getting to the specifics of this story. Of course, there would have been no use for an officers club for blacks during the 38 star flag days, as the army didn't allow "Colored" officers at that time. The 9th and 10th were rolled into the 2nd Calvary Div. in 1942, and that unit was stationed at Fort Huachuca, along with a number of other Black units. By that point, there had long been Black officers, and that's when the building in question was constructed.
So, as I stated, a building built in the 1940s has nothing to do with the WBTS.