It wasn't very difficult to find a photo of a long gun from about 1640 that had a semi-buckhorn type of rear sight.
But that still doesn't answer the question of when the full buckhorn sight was invented, and whether it's possible that a Tennessee Poor Boy rifle could have originally been made with one.
Then the OP's question is expanded to being about the particular time period of 1820's - 1830's.
While it's known this type of sight was on Hawkens, we don't know the exact time period.
I understand that some folks want to rely on precedent and surviving examples as proof of this or that.
One problem with using precedent is that it doesn;t allow for any real or imagined improvisation by any original period maker or original period owner.
And almost the whole wide world would need to be considered in order to exclusively rule out the sight as being impossible to have been correct for that type of gun in that time period.
I will freely admit that I could be wrong, that there's no way that the full buck horn sight could ever have been put on that type of rifle during that time frame.
There are only 3 or 4 possible answers, yes, no, maybe or I don't know.
It looks like TVM only made that sight standard on that model of Tennessee rifle which they clearly advertise as such before ordering.
Perhaps they intentionally use the wrong sight knowing full well that since the rifle is a reproduction, they're allowed to use some artistic license to install whatever sight that they want.
Or perhaps they feel that the sight was in use close enough to that period to justify its inclusion on that particular rifle.
Whatever their reason, they advertised that rifle as having that sight and may even give customers the option to change it before final purchase.
Quite frankly, I don't know when that sight was invented, by who or when it could have been used or installed on any long rifle for the first time.
I like everyone elses answers and now I am uncomfortable with my own.
But I don't want to simply blame TVM for not providing a period correct sight just because of their reputation either.
Maybe they know something or maybe they don't care.
But that still leaves us all without all of the facts, possibilities and probabilties.
However I do know that the customer is always right.