>> “getting a teleconverter to fit between lens and body is generally not a problem.”
In case of the Nikon teleconverters fitting is an issue. The Nikon TCs have a tab that prevents mounting on most lenses. This is because there are some TC / lens combinations that can damage the rear lens element. None of the Nikon TCs will mount with your 80-400.
AF becomes less reliable when the widest aperture is smaller than f/5.6. Putting a TC on the 80-400 give you a max aperture of f/8 – so AF may work in good light but it will be slow and not as reliable. Metering (AE) will work normally with most TCs.
The above mentioned Kenko teleconverter is the one that is most often recommended for Nikon lenses that cannot mount to a Nikon teleconverter.
I, personally, do not recommend using a TC with the 80-400. Not only is AF spotty but the image quality (sharpness, color, contrast…) can drop dramatically.
You would be better off to sell the 80-400 and get one of the xx–500mm Sigma lenses.