If it had been a large deer, of course he should have stopped, at that point it's the safety of the people inside the car.
In law for insurance purposes it needs to be clear cut, person behind is almost always at fault. But that doesn't mean they are the cause of the accident in all cases. There is nuance to these things, and part of that is that braking for a rabbit, or using Level 2 automation, is increasing the chance of an accident happing on the road.
If the way you drive increases the risk of someone driving into the back of you, even if they haven't left enough space, you are at fault in my mind.