The question is a good one. I have one reference of a bank officer keeping his 1851 Navy carried in the CW in his desk drawer to defend his bank well into 1890's. Elmer Keith, the great mid 20th century gun writer stated in his book "Hell I was there" that his first pistol was a 51 Navy and that was just prior to WWI (that is also the only reference I have for a lubed wad being placed between the ball and powder, he used an old felt cowboy hat to make the wad).
Hickock apparently had 51 Navies on him when Jack McCall smoked him in 76, but some writers claim they were conversions. The 7th Cavalry re-issued Rem. NM to Reno's troops when some of the command were sent to Arizona briefly after the 74 campaign, that is from a current treatment on Custer, the title I forgot but can get the reference for you if you desire it (the author commented that it was Custer wanting to control the use of the modern SAA).
I suppose it is
what you did, if you were using arms professionally I am hard pressed to think of an instance of cap and ball use in the 1880's. So I guess it is reflective of the persona you choose.
The picture posted by Dell is very interesting if for no other reason than the saddle - that is a looker for sure - and I appreciate his posting it for that reason alone - however I'm not sure you can read much into what surely is a staged photograph. Now if she had a Pinkerton's badge on..., I doubt she rode the range with either weapon, but I bet she road that pony better than I could!
So, can you carry your NM Army and a 92 winchester in NCOWS? Absolutely! And, have a good time doing it. As you grow into the sport you will probably want to narrow your time period, and fine tune your weapons selection to reflect that.
By the way, for my persona and year 1874 I most often shoot a NM army and did so at the NCOWS nationals this year. Came out of the factory in 1864 and is still cranking them out better than I can shoot (OK, OK..., that isn't saying much for the gun
). May yours last as long! Have a good time.