I agree that it is tough situation for AF, most notably because of the small AF target. This is something I am finding a weakness on my 5Ds R as well and I just learn to be very precise where I place the AF point.

Now even on my 1D III and 1Ds III has a significantly better time.

1Dx III, childs play. It handles situations like this like a boss.

Now for the obvious, a longer lens will fill up the frame more and make a tremendous difference and even though some of the older big whites are dated in some ways. It will be a tremendous upgrade from the 100-500mm, and depending on what you get maybe not even that much more money. A 500mm f/4 IS can be had for pretty reasonable prices or a 600mm f/4 IS for roughly $4500. And of course the veritable 800mm f/5.6 IS, which used can be had for $8000/$10000 depending on condition, age ect.

Pretty much the AF performance what you are seeing is what the R5 is capable of. You might get better hit rate with a fixed AF location, because it doens't hop around, but even then the subject is very small in the frame and as you or the subject moves is liable to lock onto the background. But it probably be better, so I would try that. As I did shoot it a fair amount R5. And it's nowhere near what it is cracked up to be. It's a 5-series, if you are coming from that perspective, then it is not so surprising. Coming from a 1-series, it's a night and day difference in more challenging situations.