Also, I'm not sure if visas, etc. were as important (or existent) in the late 18th century (correct me if I'm wrong), and therefore Ambassadors were more concerned with keeping up positive relations with the nation or state in which they were positioned. Talking out disputes before they escalated, that sort of thing.
In my "Created Equal" I have something like that because of the unique position of Louisiana - the Territory, purchased under Adams (who flips terms with Jefferson), has no slavery allowed, whereas Mississippi does. So, after a number of disagreements, the large balck population in free New Orleans pretty much requires Governor Burr to establish one across the river to address a number of concerns.
However, while it's implied that this happens in a few other states because several others do pass gradual emancipation, it's more to address a particular set of concerns - escaped slaves, kidnapping of free blacks, etc. - which ends after the War Between the States and the gradual end of slavery by 1850. (Not Civil War - if you haven't read, a variety of issues lead to several Southern states going to war in a manner similar to the Ohio/Michigan thign in 1837)
So, it's plausible to have some representatitive to another state int hat scenario, I think. The big question is, would it be possible for there to be enough issues besides that to the present day so that the Federal government stays out of it?
One thing I think would put a crimp in that is communication. It's easy to understand how local squabbles between ATL states early would be best handled that way, because by the time word gets to D.C., everything can be settled. However, once the telegraph and especially transportation of people between states become widespread, there wouldn't be as much need since D.C. could handle things much more quickly since tjhey'd learn about them much more quickly.
So, the trick would be to create that mindset of staing out of states' affairs even after technology makes it easier for the Federal government to become involved.