Eh, I don’t really agree with a lot of the advice given. First, yes, you can handhold a 500mm f/4 G. I shot the previous version of the 500mm, which was lighter, but didn’t have VR, for 7-8 years and almost never used a tripod. I currently shoot with the 600mm f/4 FL E and have never put it on a tripod. I frequently hike upwards of 2 miles with the lens on a Black Rapid strap or even just carrying it by hand. It’s not that bad—and I’m not in great shape.
Second, I think you could find a used Z8 body in your price range. But a Z8 won’t solve the problems you’ve identified. The 200-500 f/5.6 is just a slow focusing lens. A 500mm f/5.6 will focus much faster, but it won’t have any effect on bokeh. As others have mentioned, getting closer to your subject and shooting subjects that farther from the background will fix the bokeh issues. Of course, a 500mm f/4 will help, and it will focus MUCH faster than the 200-500mm f/5.6.
Third, Nikon’s policy used to be that it would repair a lens until it was two generations old. The 500mm f/4 G is one generation old, but even assuming that Nikon will stop repairing it, you’ll have some runway for repairs with businesses like APS. I’d ask Nikon what its policy is currently and whether it can repair a 500mm f/4 G. I’d also ask APS whether it anticipates being able to repair a 500mm f/4 G for a few more years.
As to upgrading the camera or lens first, I think it depends on what you’re more frustrated by. It seems to me that you have more of a lens issue than a camera issue. So, it might make sense to upgrade the lens now and save for a camera afterward. While saving, you could also work on your fieldcraft so that you can approach birds more closely and identify opportunities where the background is further from the subject, which will benefit you immensely when moving to full frame.
