Partial answer coming your way. While it is arguably distinct from the strain of sports-hunting propagated and brought into the lime-light by Theodore Roosevelt in the early twentieth century, there was a tourism industry of the nineteenth century that catered to British lords who wanted to 'escape effete over-civilization and decadence' in the 'wild' American west. At the end of the nineteenth century especially, the industry of hunting guides, dude ranches, and western nature tourism catered in no small part to second-sons of British nobility (and other Europeans, and Neo-Europes, e.g. including Australia). The nobility was not only losing its social power as the nouveau-riche merchants and pioneers of industrialization rose to power, but entering the clergy was an increasingly outdated way to dispose of extra sons and keep the estate together under the care of the eldest male. So many turned to America, trying their hand at ranching, hunting, and 'roughing it' to try and find their place and identity as powerful white males in a world of increasing social instability. Monica Rico writes that through these British male adventures:
Ultimately, western masculinity became detached form any specific place and developed into a template for hegemonic masculinity on any frontier, in any space where civilization grappled with recalcitrant colonial subjects.
She links this to Teddy Roosevelt's African hunting expeditions, in which he visited British colonial holdings (as opposed to American imperialist expansions Westward that the British visited) and shoots approximately half the wildlife in Africa (if his exaggerations are to be believed /sarcasm/) with the help of indigenous guides there. I'm not so sure on how this relates to the popular, every-man sport hunting that mid- to upper-class Americans partook in, but this is what I do know?