Yeah go the .22, you can knock Bunnies off at 100yds with a little holdover (about 10cm ), they are easy to maintain, accurate and quiet. I do all my foxing with a .22lr using win power points, out to 75yds if you shoot them in the chest when they face you they will normally drop but some do run for about 20yds or so.
Side on chest shot out to 50-75yds on a fox will see them run for a short distance then drop (its hard to run without a heart or lungs)and of course a head shot out to 100yds will drop them instantly.
Just to let you know I shot a feral cat last Saturday at around 100yds with the .22lr I got her side on just behind the shoulder and in the middle of her lungs... she turned and ran for about 75-100yds then dropped stone dead.
Save the purchase of a .223 till a bit later...if you want to eat the Bunnies shot with a .223 then a head shot IS the only way to go or past 400yds (chest shot) where the bullet has shed a lot of it explosive power.
Have you thought about a .22Magnum?, you get an extra 50yds of range and a lot more grunt, heres a .22Mag vid I made about a year ago
https://youtu.be/wFGX0_wzulg .