To the OP: Probably anything out of a 243 is going to kill em pretty dead at 300yds and further. I personally like the 87gr Vmax out of my 243 AR. I've dropped them at over 400 yards a few times -- every time I actually hit them at that range anyway. A v-max in any weight should work without blowing up the fur. The 95gr SST is another story...
For prairie dogs, the 243 is a bit spendy to shoot, but a lightweight hollow point (<65 grains) should produce some pretty 'neat' results.
As for the 22-250: I run a 53gr vmax that works nicely. Before my reloading days, I shot the cheap Winchester hollow points too. Those drop coyotes in their tracks and turn prairie dogs into a 'pink mist'.
I'll also throw in my experience with my 223: After shooting 243 and 22-250, the 223 is a huge disappointment. Coyotes tend to run off a bit instead of drop and prairie dogs tend to tip over instead of explode. Recently I've been impressed with the cheap American Eagle 50gr Varmint Tipped loads on both critters.