Also keep in mind that the M1861 was well supported by the supply system of both armies. Plenty of spare parts. The same was not true of the '63 which was new and while some parts were interchangeable with the '61 (buttplates, triggerguards, some lock parts), problems caused by heavy use could develop that would put arms out of service in a short time.
Actually, the M1863 Rifle Musket was never the front line weapon it was intended to be, many, if not most, were pulled out of storage in as new condition almost immediately after the War and converted to Allen patent breach loading arms, the predecessor to the Trapdoor Springfield. After the M1873 Trapdoor came into service, many of the '63s were surplussed on the civilian market and some were even sold to
France
for use in the Franco-Prussian War of 1871. They didn't make it in time and were confiscated by the Germans and sold on the open market through
German
retail and wholesale outfits in a fashion similar to the way Bannerman's did in this country. The M1861 is your best bet.