I think what you are looking for is called .22LR or .17HMR lol
In order to slow things down a heavier bullet is normally used rather than a light one. trying to slow things usually uses less powder which can cause a detonation rather than a burn which is a bad thing. Case fill percentages need to be kept high so you will probably have to search for a new powder.....
If the reason for a milder load is carcass damage look at bullet construction. VMax is not your friend. Look at a soft point bullet.
We used to shoot a lot of hares with a hornet and I think it was Winchester HP type ammo. We got a lot of carcass damage. By chance we ran out of ammo and could only get S&B soft points. Carcass damage was reduced so much we never went back to Winchester.
Mark