But then they would be prosecuted and punished as well as having their animal exterminated. As I said, it needs the full weight of the law behind it making it non-optional.
When hand guns were outlawed in the UK, people handed them over as they had no choice. Yes, some wouldn't have complied no doubt, but while they might have been able to hide a gun away, they'd know they'd be in serious trouble if it was ever found with a prison sentence very likely.
Same thing should apply to these dogs, and you can hardly hide one of those things away to the extent that no one would ever see/hear it and report it.