Yes, your choice of the 223 is fine, you mention casual shoot, as a diversion from deer or antelope hunting.
I've settled on the 223 & 22-250, nothing wrong with the 204, well, besides sorting the empty brass. I the platform is a biggie. My most capable gun is a custom 22-250, for dispatching dogs, it's tops. My reloads with it are modest for a 22-250, still more than any 223.
Over the years I've usually gone mid June, lots of 1/2 size dogs. With that, the bulk of our shooting has been 50-200 yards, spoils a guy. I've used down to 'sporter weight' rifles.
Of course a clear 14/16 on up power scope with a good trigger factor in. My neighbor often goes to WY for antelope, I always tell him he should take a smaller rifle for dogging after the tags are filled.