Given how far out it is you can hardly say it's even part of the solar system. I agree with the assessment that it should be in a class of it's own.
I believe as long it's in the influence of the sun, it can be considered part of the solar system. I agree it will be a different classification, just like how you described. But it's not the only planet that is beyond Pluto. We have 2, more planets in our solar system that is the classification of Pluto. So far without including Tyche, we have 11 planetary bodies, (planets and sub-planets or what ever they decide to call it today) and I believe we have even more.
Just checked we have more. Sedna, Quaoar, and new ones I didn't know about, Orcus and 2007 Or with little number 10 beside it, Makemake and Hauemea, so that is 13, and 14 if Tyche is out there. Doesn't matter if he originated in our solar system, he resides with us now, so he is part of the family, I guess you can say he was adopted. So still part of the family if true.

So I lernt something more, we have way more than 13 planets in our solar system.
It's amazing what they call the border of our solar system. I didn't know much until I started looking into New Horizon. I thought our solar system was just the end of Pluto. I am shocked that is like only the 1/2 way point or something like that. We have the Oort cloud, Kuiper Belt, the we have the Bow shock and a few other names I can't remember. Voyager I and II are still in the influence of our sun so they are not in Interstellar Space yet. I am surprised we have no pictures of the Oort Cloud and beyond. Just goes to show you how vast it is out there, that Voyager I and II don't have anything to show for being so far out now.
*edit* Proper name for Pluto and planets like Pluto are called Dwarf Planets. Sorry for any confusion.