If what you say is true, do not pay them and let them take you to court. They left the scene of... well, a hit and run, and that seems v-illegal and shady to me. Who knows what they hit before or after.
Is your dog big?
There is no way they can prove their claim, besides it sounds like they were speeding?
If I was a judge, I would laugh this out of court.
if it goes thru I would counter sue for wreckless endangerment of the neighborhood, and hit and run, since they left the scene of a crime.
BTW: do you have homeowners? you might want to discuss it with your agent.