Pictures help but the "P" is usually there as a proof mark put on at Springfield Armory to show that the barrel passed pressure testing . If the barrel is 22 1/2 inches, it probably is, then, a cut down Krag rifle barrel. That's real common. Long 30 inch rifle barrels were frequently cut down to be used and carried easier for hunting. The .30-40 Krag round is a great hunting caliber in this part of the world.

I don't have my lists with me but the Krag serial number 81502 was made around October 1897.

By memory, I believe that this was made slightly after the second grouping of M1896 carbines and would, therefore, have been originally made as a rifle. That would confirm that this was a M1896 Krag rifle cut down to carbine length.