Hare weight at maturity is about 3 kg or maybe bit more and cottontail is something about 2 1/2 to 3 pounds if I'm right. So there is a quite big difference of sizes. Big brown hare duold weight 6 kg.
In my opinion ban of a 22lr is futile but no can do. I do also have more powerful but still handy rifles available if I need one for hares