Luckily neither of us were hurt (nor my cousin's dog) and the Mini seems to have fared pretty well. Despite coasting to a stop after about 50 feet, the damage seems to be limited to the brake disc, lower control arm and my spirits. Maybe this is the sign that I need to start tearing down the whole car.
The upside is that we were test driving it and were then planning on pulling the exhaust out after it had cooled off. While coming to a stop, the car spent some time riding on the mid-pipe so I guess it's a good thing we didn't have the new exhaust on.
In the end, we called AAA to tow us. We got the wheel back on, but figured it was safer to get towed than it was to risk driving the car until we knew what kind of damage we were dealing with.