Hi. I do have a 200-500 and an AF-S TC-20E III. I also have an A-FS TC-14E III. The 2X TC pushes the f stop to f:11. The AF on my D850 or Z 7 don't work above f:8.
I have tried manual focusing with the 2X TC on a tripod but at 1000mm any slight vibration kills the shot. I recommend either setting the shutter delay to about 2 seconds or use a remote shutter release. You might also want to lock the mirror up before the shot. The 14E III pushes the 500 to 700mm at f:8 where the AF still works. I could never get my focusing to be as pin sharp as the 200-500 is capable of, so I don't use the TCs with this lens. I close crop the 45+mp images. The super high resolution of my Nikons is almost as good as, and much cheaper than, verrrry expensive glass.
I do most of my bird shots with the 200-500 in Manual, hand held at 1000 sec at f:5.6. You can see some of my shots at PhotoArt by Walter Blady. I should also mention that these on-line jpegs were saved at about 6o% quality to save download time. The actual hi-res shots are much sharper.