I think you are right about IQ of the D810 in the DX crop mode.
I do not shoot crop modes with my D800E, because I do not believe in pre-crop a photo. Memory cards are quite inexpensive and many are very fast, so I always shoot the full frame and then crop at post. This way I may crop off 10% (32 MP), 20% (28 MP), or even 40% (21MP) as needed. Most of my crops were off-centered, making DX crop mode less practical for me.
As I had some very talented and devoted guides in Africa, and perhaps African games are more accustomed to people, I was not often focal length limited, even with just a Tamron 70-300 VC lens. There were quite a few times when I had to back down the zoom to 100 mm - 200 mm range on elephant, white rhino, and cape buffalo.
The D500 with its advanced AF system should serve you better than the old 15 cross type focus points AF system D810 uses. Higher frame rate also helps in certain situations. Since you do not own either a D810 or a D500, choosing to rent the D500 with the reasons you mentioned seems a very logical decision.
Good luck and enjoy!