I use a 223 for my varmint. I wanted 204 but got a 223 as the price was one I could not say no to.
For dedicated varminting I would still go for a 204. I have seen some amazing shots with the 204 and am very impressed.
Tread carefully if you go 223 or 22/250. Nothing wrong with the calibres. Both do the job well. 223 I use a 1:12 twist. I tried 1:7 so I could use heavier bullets but it struggled to stabilise the 50gr and 40gr varmint pills. If you want to shoot the odd deer etc with it the varmint pills wont work so well. (or not at all) If varminting and you want fast flat(ter) rounds then lighter pills through a 1:12 or similar should give you good results. You say in your original post you want a dedicated Varmint rifle so I would suggest not considering the faster 1:7 type twists on the 22 cals.If it is a dedicated varmint rig then you possibly wont want heavier pills in it anyway.
I have seen varmint rigs in 243 as well as 6.5x55 but I would still go 204 for my own choice if I was looking again.
Decent glass will make a bigger difference than calibre choice. For rabbits out to 400m either 6-24 or even at a stretch 8-32 would be my suggestion but its probably getting a bit big.
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