There wasn't a Tennessee until 1796, except maybe a Tennessee River Valley or a Tennessee County, NC. The populated part of the state was either part of NC, part of VA, the State of Franklin, or the Southwestern Territories before the State of Tennessee was formed.
I could easily be wrong on this, but I don't think the name Tennessee Rifle was applied before Turner Kirkland and other fairly recent collectors started to describe the iron mounted rifles that way. So the answer to the OP's question:
So, what would be the approximate date that the rifle would have been recognized as the Tennessee rifle.
may very well be
post WW II, as nobody called them that before then. I could be off by a few years or decades, though.
Other possible questions might be
"When were rifles first built in the area now known as the State of TN?"
"When were rifles first built in the State of Tennessee?"
"When were iron bound rifles that look like what we now think of as Tennessee Rifles, first built for use in what we now know as the State of Tennessee?"
These questions may all have different answers.