I know that the 17 HMR with non toxic ammo is being used widely by the likes of Forestry commission and other agencies for vermin control and has replace the 22lr.
I have a Remington 700 in 223 with a 1 in 12” twist rate. It shoots a 50gr copper bullet very well. Below is the effect on a crow at 150 odd yards after being hit by a 51gn Peregrine bullet. Same bullet works very well on Roe, and whilst I haven’t yet I suspect effect on a fox will be equally terminal.
223 is easy to get ammo for with plenty of non toxic options. And I would probably go for a 1 in 9” or 10” twist as that gives you a greater choice of bullets. 222, in older models tends ti have a bit slower twist rate, yet I know
@Edinburgh Rifles have developed loads that work well in older CZ 222.
I am waiting for a 1 in 12” twist to line a Rook rifle to 22 Hornet to give me a light small game / vermin rifle.
Will probably drop down to a 35 to 40 odd grain bullet of non toxic variety for this application.