I am torn in the middle of your situation! I am a mother of a 6 year old with low self esteem, and a very well adjusted 8 year old.
I am trying my best to help my 6 year old threw what he is dealing with. Never have either one of my children harmed a animal! My oldest saw a dead bird at 3 years old and demand a explanation!. We currently have a 5 month old yorkie and this puppy has been a blessing! Our puppy does not feel safe unless she is touching someone! Our kids have even noticed this trait!
Kills who harm animals have been proven to have many social issues. I would have them checked out ASAP regardless how embarrassing it is to you. It is a very easy lesson for kids to understand that all lfe forms need air to live, you kids may need some extra help in this area!