Strictly speaking, my guess is that the tenant is responsible as their actions caused the damage. If they fail to pay, then it’s down to you as the landlord, since you are responsible for their actions. Assuming you don’t have landlord insurance, strata insurance might pay if you refuse and the tenant can’t or won’t pay.
As for the increased cost, it may be due to the fact that this is a one-off, or because the company replacing the door thinks it will be an insurance job and therefore bump up the price. It’s certainly worth questioning.