First ... A CAVEAT
I wasn't there. Nope. Can't speak from experience. Wasn't present. No Holiday Inn. So don't quote me. Much.
At the time of the Span Am conflict, neither Colt nor Smith & Wesson or anyone else was proofing for nor warranting their respective handguns for Smokeless Powder. That didn't happen until after 1900 and later. Also, the military procurement system then, as now, procured ammunition in quantity (a bazillion rounds at a time) and retained it in stock until it was used up or exceeded it's shelf life (arbitrary limit) and was surpluse'd.
Ergo. The military had procured and retained on hand several rounds of BP ammunition. There were no guns proofed for Smokeless at the time and the military would have simply issued what was on hand. BP was on hand. Lots of BP was on hand. I would therefore, expect the only "issue" handgun ammunition to have been BP.
Back to the CAVEAT: Wasn't there. Don't really know. At this point in time, I doubt there is anyone whom can definitively state the issue "handgun" ammunition was "this" or "that." The logistics, however, would guide me to BP. Your guess is as good as mine.