Having had our cat run over last year, know how it feels, unfortunately he was too injured and the vets advice was to let him go. Hope your pup pulls through, I can understand how you mean about going to him despite being bitten, I'd do the same if it happened to my dog, as I'm sure any dog owner would.
How fast was the driver going? 10yrs ago I was unfortunate enough to hit a dog, totally unavoidable, A14 just past Bury St Eds, stray dog ran from 1 side, jumped and cleared the barrier, straight into the front of the car. I stopped, luckily was about 11pm on a Sunday, nothing much on the road, ran back to where I hit him, found him still alive, carried him back to the car but he died in my arms. Had to leave him on the side of the road, had my (then) pup in the car, only 8wks old. Reported it to the police, and let the insurance company deal with the damage to the car - smashed bumper, washer bottle, fog light, kerbed wheels.