It would not be a Hawken.
Let me explain my thoughts.
Prior to the Lewis and Clark expedition most of the Bison east of the Mississippi had already been killed and that was with flintlocks, at that time mostly full stocks.
When Lewis and Clark returned in 1806 they met with boats and hunter/traders going up river which again would have been obviously flintlocks and could have included some 1803 contract rifles.
So from 1806 to arguably 1836 all the rifles going in to the west would have been flintlocks, 30 years of killing buffalo with flintlocks before percussion guns even made the scene. How fast hunters gave up their tried and true flintlocks for new technology percussion guns is up for speculation but I am also one of those who believe that when they did there were more Leman's and others than genuine Hawken's. Rifles were being made by lots of less famous makers and many in larger numbers than Hawken.
Percussion guns then made the scene but in a very short amount of time were supplanted by cartridge rifles. Remington introduced the rolling block in 1867.
I believe the name Hawken is similar to the name Sharps, there is no question that there were many many more Remington rolling blocks on the frontier in the late 1800's than Sharps rifles (factory records show production numbers of rolling blocks dwarf the number of Sharps produced) and no question more bison were killed by rolling blocks but the Sharps has "the name" (makes you wonder what would have happened if Matthew Quigley carried a rolling block).
It is a little late for the time period I am interested in but I think if I ever go buffalo hunting out west again I will use a 1803, seems like the perfect buffalo gun, at least until the rolling block came along.