We have a dog that a trainer told us was untrainable. His stupidity is confined to less annoying behaviors than killing neighbors’ cats, though. Not sure of a solution for you, but I do know that some dogs just don’t learn.
Do YOU have a fenced in yard? I didn’t notice if you said you did.
If not, get one! Get one big enough that can’t be jumped.
That’s the only thing that will 100% work. IMHO
I personally detest electric fences. Dog may or may not get out, but it does NOT keep anything else from coming in!