@Matthew1514 IMO, for this amount of damage, use your own insurance if you have a policy that covers damage by renters/guests. If you can obtain the name, address and phone number of the guest who did the damage, provide that information to your insurance company or provide the AirBnB reservation facts -- your insurance company may be able to get reimbursed by AirBnB.
You can still try to get AirBnB to pay, but I wouldn't hold my breath. They will probably discount the garage door by the length of time that it has been in service, i.e., when it was first installed.