Sorry, but the OP was in reference to BP handguns, as I recall, and LEOSA in this day and age would likely not apply. The privilege to carry concealed only applies to the individual's firearm of a type that their dept./agency issued/authorized, and provided or allowed a qualification course for.
I'm not too familiar with the qual. courses other agencies are running, but I believe the time constraints, mag. change/speed loader drills included in ours would kinda' negate most of us from qualifying, even if we could find a CLEO and RO to sign-off on it.
The main question was if NY laws address his 1860 Army revolver. The short answer is a definite "YES", once also in posession of components necessary to fire it.