High refresh rates on low resolutions are more cpu dependent, whereas higher resolutions on lower refresh rates require more gpu power. Just like how everyone else is saying, I highly suggest doing a bios update and snagging the 5600x or 5800x, as they are all 6+ cores with fantastic speeds, ipc, and low latency. I might even recommend the 5600x, I've never heard of a game where "6 cores with multithreading isn't enough" and the single core benchmarks are very promising.
I would take into account the different prices of the 5600x and 5800x, relay that to how they impact your budget while considering the performance difference, and pick one accordingly.
If you HAVE to get a Zen 2 chip instead of Zen 3, I'd actually suggest the 3800xt. The price is often high, but it dips every now and then. A couple tech outlets on youtube did testing and concluded that this chip had the best single core speeds out of the other Zen 2 chips, ultimately leading to the best fps on your 1080p 240hz monitor.