Love groundhog shooting.
Now I seldom get the chance to do much, but over the years I shot quite a few. Of the three guns you mention, to me the .22-250 is a dandy. I have not used a 204, if I were in country that I could still shoot chucks, I would probably own one.
For general chuck shooting, it is tough to beat the old .22-250. As far as brass life, Ladobe nailed it down. If you are pushing your loads to the top, brass life is short. I neck size, and seldom loose many cases. Once they are trimmed a bit short, and if you neck size, you brass life should be fine. As far as the 204, with the new powders and 40 grain plastic tipped bullets, I wonder if the .204 really has much on the .22-250
One can drive those 40 grain very fast, and plus have the option of the heavy 55 grain for longer shots or wind. But from my chuch hunting days, wind was not serious as it is for coyote hunting. But to be a REALLY well equipped chuck hunter, I suppose you should have ALL THREE PLUS A 22 Hornet.