I shoot 204 Ruger/223/22-250/6mm-284/6.8 SPC for gophers/predators...
If I was out for gophers more than anything else it would be a toss up between the 204 Ruger & 223 I rarely even consider taking the 22-250/6mm-284 for gophers these are reserved more for coyotes/wolves etc.
I haven't taken the time to check the twist of my 204 Ruger barrel yet it is a 23" bull barrel I load 32gr V-Max @ 4040fps my rifle is a T/C Contender carbine that I switch barrels on.
One of my 223's is another T/C Contender carbine barrel it is a 21" tapered 1-12" twist rate I load 40gr V-Max @ 3400fps both of these barrels/cartridges/loads consistantly give me -2" 300 yard groups.
Cost to reload once the brass is purchased are almost exactly the same on any of these cartridges so that isn't an issue for me.
I'm now also shooting a Robinson Arms XCR-L with uppers in both 1-9" twist 223 & 1-11" twist 6.8spc both are going with me tomorrow to Alberta for the 223 I'm taking the exact same 40gr V-Max loads but velocity is appr 3200fps - 3250fps in the shorter barrel, as well as 50gr Blitz @ 2900fps and 60gr V-Max @ 2950fps - 3000fps and the 6.8spc will be loaded with 95gr TTSX @ 2825fps and I would not hesitate to shoot out to 450 yards on coyotes with the 95gr TTSX loads...
I understand that the 6.8SPC is available in a Rem model 7 this should make an excellent gopher/predator round combo and I like this cartridge so much that the first chance I get I am adding a T/C Contender barrel in 6.8spc to the stable...
What you say you are looking at doing go with either the 223 or 204 both will do anything you would ever need the 204 will just do it with a far flatter tragectory and I do not shoot thise beyond 300 yards.

If I was out for gophers more than anything else it would be a toss up between the 204 Ruger & 223 I rarely even consider taking the 22-250/6mm-284 for gophers these are reserved more for coyotes/wolves etc.
I haven't taken the time to check the twist of my 204 Ruger barrel yet it is a 23" bull barrel I load 32gr V-Max @ 4040fps my rifle is a T/C Contender carbine that I switch barrels on.
One of my 223's is another T/C Contender carbine barrel it is a 21" tapered 1-12" twist rate I load 40gr V-Max @ 3400fps both of these barrels/cartridges/loads consistantly give me -2" 300 yard groups.
Cost to reload once the brass is purchased are almost exactly the same on any of these cartridges so that isn't an issue for me.
I'm now also shooting a Robinson Arms XCR-L with uppers in both 1-9" twist 223 & 1-11" twist 6.8spc both are going with me tomorrow to Alberta for the 223 I'm taking the exact same 40gr V-Max loads but velocity is appr 3200fps - 3250fps in the shorter barrel, as well as 50gr Blitz @ 2900fps and 60gr V-Max @ 2950fps - 3000fps and the 6.8spc will be loaded with 95gr TTSX @ 2825fps and I would not hesitate to shoot out to 450 yards on coyotes with the 95gr TTSX loads...
I understand that the 6.8SPC is available in a Rem model 7 this should make an excellent gopher/predator round combo and I like this cartridge so much that the first chance I get I am adding a T/C Contender barrel in 6.8spc to the stable...
What you say you are looking at doing go with either the 223 or 204 both will do anything you would ever need the 204 will just do it with a far flatter tragectory and I do not shoot thise beyond 300 yards.