I know everyone at the City will hate this idea, but it has been done in many, many cities across the country.
Can we disband the department and turn operations over to private groups?
In these other cities, they send out requests for proposals for defined services at a fixed budget.
Performance levels can be fixed and if the private group cannot meet their charged goals, they would lose the contract.
It could be done in any number of ways. One way is to have each shelter run by a different private group or cooperative of groups. Each shelter would have a separate budget determined by the usage and types and conditions of animal brought in.
All groups could be separate or there can be an oversight central command structure.
Since the dept would be disbanded, employees would be shifted to other departments, such as public relations or to the mayoral staff. Good employees would be rehired by the shelters.
Just an idea. I don't know if or how it could be implemented.