Unfortunately energy retained within the body is too small to have any meaningful effect. The only thing that matters is the wound tract and what that destroys.
I had a squirrel once take five .22 pellets in the body from a Webley MKIII probably running at close to 11 ft/bls and as it was directly up a tree no more than 15m. Four of the 5 pellets found in the body when skinned out. There was enough life left in the squirrel for it to run down the tree and leap at me
It landed dead at my feet.
Rabbits are even bigger and unless heart or brain destroyed can easily take all the energy from a 12 ft/lbs rifle.
Beyond 25m .22 can feather plug in wood pigeon, though a .22 is very effective at wing route shot at close range barn pigeon shooting.
You say a couple of mm off perfect. Well most people can't drive a .22 a couple of mm off what they can with a flatter shooting, and better behaved .177. The real issue is are they shooting polo mint accurately enough?
The crux for ethical air rifle shooting today is accuracy with perfect shot placement. Make a poor shot of it and bigger calibres have nothing to make up for that error, and they are just that more difficult to drop judge perfectly. Shoot perfect with any of the calibres and they should bring in the bacon.
I actually prefer .22 for rats, but then I'm not testing the range at all. For opportune walk about then it is accuracy, and the .177. Be familiar with a good shooting .20 combo and that works fine too. PCP means anyone should be able to manage some really useful consistent accuracy.
Add 6 to 12 ft/lbs of energy, FAC, then everything changes.