In the period this took place it would be unusual for a woman to be allowed to be a bounty hunter. At this time women weren't allowed to have bank accounts, sign legal documents, hold office or even vote. This premise of a woman bounty hunter in the 1840/50s is incorrect.
When Will Cotrell is shot during a fight he has his back to the shooter while he is on the ground on top of his opponent. When he gets up he indicates that he was shot in his shoulder right at the upper arm joint. Later he is holding his arm just above the elbow joint and there is no wound to the shoulder area or the elbow area.