The point being the government could make an argument for an exigent need, but that doesn’t relieve them of the their obligations to follow both due process and to conpenastion requirements.
The applicable due process is to be judged dependent upon the circumstances. The cops had no time nor requirement to get a warrant.
Damages? Yes, fix the door, although the damage to the door is the sole and direct result of the homeowner’s refusal to open it.
A smart city would give him a new door. The driving problem here is that all the damages were caused by the homeowner’s refusals, not the LEO’s legal needs nor actions.
Had the homeowner simply opened the door and made his legal arguments later, none of this would have happened. Any legal arguments would have failed, however, as the LEOs had a legal right to entry under an exigent circumstances provision.