I'm not sure I'd bother with a TC, to be honest. Even with as excellent a lens as the Nikkor 300mm f/4D AF-S, you're losing a little sharpness to the TC, and as high resolution as the D7200's sensor is, I'm not confident (having tried such a combination for several months) that it pays off versus simply cropping, with the additional downside of the TC dropping the aperture to f/5.6 - with British weather as it so often is, I tend to find myself thankful to be at f/4.
As for the original question: yes, a D500 is the best crop format camera Nikon makes, and a top-notch performer, by all accounts. However, I'd concur with the sentiment expressed by others - whilst the AF is supposedly All That, and the continuous shooting rate even higher than the D7x00 family, there's a sharp difference in price between, say, a used D7100 and a new D500.
As for the lens - there are a few very good options, including Sigma and Tamron's zooms, and Nikon's 300mm. As regards the latter, bear in mind there are two versions you'll commonly encounter - the 300mm f/4D AF-S, and the newer 300mm AF-S f/4E PF ED VR. The latter comes with stabilisation, which may or may not be useful for you, and is half the weight and length. I use the former, which can be found used for around the £500 mark from the likes of MPB. If the budget permits, you might want to try a quick 3-day rental of a couple candidate lenses, and see how it all works out for you, with AF speed, sharpness, and weight.