I feel you may have answered your own question, by saying the A9III is too low a resolution so for that reason its off the table.
If you ever want to shoot in DX to gain the perspective of longer reach, its at a huge compromise as the file sizes are smaller by comparison to a higher resolution sensor.
The A9III is a very different tool.
I would wait for the Sony A1 II if anything. Or go the A1.
The A1 is absolutely more than capable in the focusing department to meet all your needs.
Again i would wait for the A 1 II as there will be significant improvements on what is already a proven thoroughbred.
I feel dynamic range and resolution is very important, especially if your cropping at anytime, or shooting in DX mode which at times is very opportune, people often say you shouldn't crop, i disagree, crop if you have to or want to, filling the frame with the right glass is the first choice of course, but sometimes you cant.
While the 200-600 is a very nice lens, You will get nicer results with better glass, especially the bokah.
Rent before you buy has its merits.
Good Glass technique and light has far better results than just upgrading the body.
Short answer
I would go
A1
A1II
and work towards better glass.
Oh i am a Nikon Shooter by the way but have the up most respect for teh Sony gear.
If you want to experience stunning tracking you need to rent some Cannon tools, in my book it has the edge over the other two.
Only an opinion