Legal Question in Criminal Law in California

You can not sue anyone's 'insurance', you can only sue the person responsible, and then only if you have a valid damage claim. If they want to invoke their insurance coverage, if any, it is up to them.

For an animal, the reasonable replacement cost [market value] of the animal is the most you could recover if you win in Small Claims Court. There is no recovery for 'emotional distress' no matter how enamored you were with the animal.

To win in court you must PROVE with admissible evidence what person caused you what damage. You said it was hit and run. How are you even going to locate and prove who did this? Did you see and get a photo of the license plate and the driver and the damage to the vehicle? Did you file a police report at the time?

You must first attempt to settle your claim with the driver, for the replacement cost. That attempt is what the Small Claims Court requires before you are even allowed to file the case, and you will be asked on the court forms what you did.