I was ready to say 223, until you said "bobcat". Hmmm, dunno now. Are you after pelts for resale, or just after a cat for your own trophy room? Gotta watch for those exit holes if you're cashing in pelts. Maybe the 223 but 40 gr bullets instead? Or a 221 fireball? I still get exits with my 223 and 50 gr vmax about half the time on 'yotes, unless distance is over 100-150 yards. I'd be afraid of blowing a hole out of a nice cat pelt if one came in close before I got him.
I don't have a 204, but my target shooting buddy does. He's never varminted with it (pity), so sorry I can't comment on that one. I would think a 22-250 would be too much of a good thing on cats. Maybe real good on 'yotes if your ground is wide open and shots are long. Where I've been calling the shots have been pretty close, usually less than 100 yds, and that would make a mess of things...
I'm not so concerned with it in the summer months because the pelts out here are all junk by then. 120 degree heat equals junk fur.
I just picked up a 17 Rem bbl for my Contender from Ladobe (thanks Larry!), I'm hoping that becomes my "go to" bbl for pelts. Still rounding up components and dies for it though.