Yes, sadly, the Fire TV - with or without you - must head north to the other Carolina to be paired. Note that if it is the latest generation Fire TV (“4K Ultra”, with a dongle) there is a hiccup in playing Tablo which requires a new firmware from Amazon, eagerly awaited but with no sign of it. (See this thread.)
However, if it is the earlier generation Fire TV (a box, not a dongle) or a Fire Stick it will not only work well with Tablo Connect, but I have found seems rock solid and never loses pairing (once done).
I don’t know about the TVs with Fire OS built in or newly planned in collaboration with Best Buy. These could have the working-well operating system/firmware or the newer, problematic software.
So, check before you make the trek to go pair, but if you have a workable Fire device, Remote Connect works great with it. I’m looking at a U.S. television program with my Fire Stick in the UK right now as I type away.