The old adage is .177 for feather, .22 for fur.
Probably because .177 would fid it easier to penetrate through the (armour) feathers of a bird.
and .22 has more weight /shock for the larger animal (rabbit).
I think pointed pellets are better for getting between the feathers of birds, but correct me if I'm wrong
... I'm sure someone will ... sigh ! ! !