So you still want to offer veterinary clients a discount?
If you still want to offer an incentive to get new clients in the door, try implementing a "soft discount." Offer a product or service instead of a cash discount to provide an incentive to pet owners—complimentary grooming with your professional groomer or a day of luxury boarding in your boarding facility. If you don't offer those services, my bet is that the groomer or boarding facility you normally refer to would be happy to work out some mutually beneficial arrangement with you. Other rewards you could offer might be a complimentary pet ID tag or a free month of flea control or heartworm medication.
The idea is to offer a product or service that your practice provides that has a high perceived value, but which might not cost you very much in the end. Plus, it may entice the pet owner to start using those products or services within your practice.