So because ones plate is full, we decide to end a life? Do you know how many dogs die a day because life happens and plates are full?
The OP has stated this dog is actually fine around animals when supervised, it's when he is not supervised that there is trouble. Sounds like he is entertaining himself. There is no boundaries. Let him out to go to the bathroom, bring him back in. He doesn't get freedom if he doesn't behave. The problems started when he got freedom he wasn't ready for. So instead of jumping to euthansia, wouldn't just supervising the dog be the next logical step? How about finding a home without kids or dogs?
The dog is going to pay with his life because yes the plates full and he no longer fits the lifestyle. It's that simple. Dogs are not disposable. They are a life long commitment. When things change, it's up to us to do right by them. There are lots of options here, starting with giving this dog boundaries and way way less freedom to do what he wants.