The D800e and D810 are similar in image quality but the AF is a little better on the D810 and it has a revised inner case frame that was changed to strengthen the bottom frame that had cracked on a number of dropped D800. For sports and wildlife, the D810 might be worth the difference.
Both are pro quality with good weather protection like the D500 and D5, and a higher grade of components. If you shoot burst, the D500 is a better wildlife camera because of the narrower field of view on all your lenses, that appears to be more reach. The D7200 is a fine general purpose camera that meets the needs of 90% of the photographer's subject matter. It is not as rugged or long lasting as the D8x0 series, has lower dynamic range, less able and slower AF, and fewer options in custom settings. For wildlife, the D500 is hard to beat because it is optimized for speed and more hostile environments. If you were doing portraits or landscape, by all means, D800, 810, 850, Z7 are better choices.
I never suggest anyone change cameras unless they have very specific restrictions on their current camera. If it is a vague feeling that a later tech or more pro-level camera is going to produce better results than one might be disappointed. Almost all "deficiencies" on cameras designed in the last 10 years really come down to technique or esthetic problems with an image. If you go to a photo gallery and see large prints, seldom is the equipment mentioned but you would not be able to tell which camera took any of them, but you might be able to identify film vs digital. Composition, vision, light/shadows subject, etc are all far more important in an imaging being appreciated. The best investment for a wildlife shooter is a workshop or mentor who had mastered fieldcraft....how to find and approach subjects.
Wildlife is getting harder to find most places in the world due to loss of habitat so skill in approaching or anticipating the subject's behavior becomes more important by the year. 50% of all mammal species that existed in 1950 are now extinct. In 30 years virtually none will be in the wilds except for some regions with vast habitat and no people like central or eastern Russia and some isolated areas of Brazil. So anything you can capture an image with now will be far better than any camera in 10-30 years.