Will God punish whoever kills a sick cat after a veterinarian advises killing it, and what can be done to absolve oneself of this sin?

May Allah reward your efforts in treating the cat, for kindness to animals is a praiseworthy quality. Some scholars have mentioned that it is recommended to slaughter an animal that is unlawful to eat if it is on the verge of death, as the author of Al-Kafaf said: "And slaughtering that which is unlawful [to eat] if it is near [death] benefits four; have mercy and you will be shown mercy." If the veterinarian believes that the illness will lead to its death, then there is no harm in expediting its demise.