I would have thought the main reason for owning a D800e is to get top optical quality.
While the 28-300 is a good all round lens for the money it would definitely not be my first choice for top optical quality at 300mm on a D800e, and less so with a converter.
That said if it is all your budget currently allows my experience is,
with good long lens technique you should get less loss of quality with a converter than cropping the D800e file size to 9 MP - which is the cropping equivalent of using a 2x converter.
http://www.naturephotographers.net/ejp0801-1.html
Nikon AF converters do not fit so you are restricted to Kenko which, although good for the money, are generally accepted as not quite the equal of Nikon.
Nikon's MTF show the recent 80-400 AF-s at 400mm as slightly ahead of the 70-300 VR at 300 mm. My 80-400 AF-s at 400 mm without a converter gives better image quality compared to the 70-300 with Kenko 1.4.
The 80-400 AF-s has better VR and focusses faster - but with the 14e to get 560 mm equivalent costs about 10 times more than a Kenko 2x for 600mm equivalent your current zoom.