Charge pet owners an initial pet fee to own an animal. So the responsible ones will help subsidize those who don’t register but enough will and use this money to pay for shelters. Have an annual smaller renewal fee. So you pay a one time starter fee (higher cost) and an annual renewal fee. Same price regardless how many animals. List the type pets that need it (dogs/cats) but you only pay it once. So if you buy a dog in Jan you pay X and if you buy a cat in Jun no charge since you have a permit. Next Jan pay a smaller annual renewal fee. All can be done online.

This way pet owners are paying for pet problems.