Thank you for your question. I can certainly understand why you feel some concern about your son's behavior. It sounds like you have tried some age appropriate techniques and I would encourage you to continue to use them. An immediate intervention for his own safety and for animals is to either supervise all contact with animals, or either limit the contact or stop it until the behavior changes.
Children between two and four years of age show aggressive outbursts such as temper tantrums and hurting others or damaging toys and furniture because they are frustrated. This behavior can escalate to animals and may be an indication that the level of frustrating has intensified and he is struggling with managing the intensity of the feelings. Children may go through a brief period of aggressive behavior if they are worried, tired, or stressed. If the behavior continues for more than a few weeks, parents should talk to pediatrician to rule out an physical problems that result in feeling frustrated and also consider taking them to therapy. Therapy can help children learn appropriate ways of managing their feelings. If the aggression becomes a daily pattern for more than three to six months, it could be a serious problem.
Parents can control the aggressive child is various ways. For example, you should intervene quickly but calmly to interrupt the aggression and prevent the child from hurting another person or animal. Younger children may need a time-out to calm down and before rejoining a group(one minute per age). You may need to repeat the time-out many times to give the message that you will not tolerate this behavior. It can be very tedious but its important to not surrender to your own frustration or buy into the notion that it is not working. Children will test limits which means poor behavior or refusal of accepting consequences could last for hours, days or even weeks. However, if you consistently provide the consequence they will eventually accept them and behavior will change.
Simple rules about appropriate behavior are easier for a child to understand than lengthy explanations. For example, we do not hurt animals rather than explaining why that is so important. Parents can affirm feelings while stressing that all feeling cannot be acted upon. "I know you are angry but its not okay to hurt animals, please use your words and tell me how you feel."
It is important to remember that parents should not expect the aggressive child to be reasonable when he or she is upset. The child may need time to calm down and may need adult support. Encourage you child to come to you when he is upset, hopefully, before violence occurs.
Finally, consider therapy if the behavior continues despite your intervention and if his aggressive behavior intensifies. Its important to get support from a professional to ensure that this behavior does not get reinforced and becomes a serious problem later.
I hope this information is helpful. Best wishes to you and your family.