I don't think you can make a definitive conclusion based upon that diary entry

The gun was loaded with twelve balls. That, in itself, does not tell you anything more than the gun was capable of holding at least twelve balls. It goes on to say that the rate of fire was twenty-two balls per minute. That, also, does not say that it had a magazine capacity of more or less than twenty-two balls.

There is simply NOT enough evidence that Beeman's gun is THE one. I could find an 1873 Springfield Trapdoor carbine within a certain serial number range, conjecture that the dings and scratches, ornamentation and so forth were the result of its being at the Battle of the Little Big Horn and subsequently during its use by the Sioux warrior who picked it up. However, to museums and serious collectors, I could make no greater a statement than the gun might, just might, have been a Custer gun, and that due solely to the serial number. Barring some form of solid documentation, that is ALL it would ever be. The L&C gun falls into EXACTLY the same category. Quite frankly, it's wishful thinking.