I'll assume it is legal to do this where you are. No need to take that any further.
You'll get the least amount of meat damage with the classic 158 roundnose lead bullet. The very reason it makes a terrible self-defense bullet is what makes it a great small game bullet: it passes straight through with little tissue disruption. Neither grouse nor rabbits are very tenacious of life, and the ones I've shot with this load just fall over in place.
Any semi-wadcutter or full wadcutter will make a mess. BTDT.
