My girl Maddie never caught a rabbit but she did make a deal with them when she got older. See here.
dog made a deal with the rabbits
I've seen plenty of dogs eat wild game such as rabbits, chipmunks, and squirrels and it didn't seem to have any ill effects. I am still reading up on raw diets but fresh live game doesn't seem much different to me than a raw diet.
Maddie once caught a crow. She crested a hill and pounced on the crow before it even saw her. But she really didn't know what to do next. So she looked at me while standing on the crow. I gave her the OUT! command and she backed off letting the crow fly away. She was such a good girl.
For many years after that all the crows cleared the park when we arrived and then starting yelling at us. Crows are pretty smart.