There should be an insurance requirement to register regardless of breed. Unregistered dogs of a certain age would be subject to "towing" just as an unregistered car would be. Give the owner a certain period of time to comply, and after that, preferably find a foster home... If that's not possible, unfortunately the dog would pay the price for the owner's negligence (which pretty much already happens, anyway).
Some breeds would require a higher or lower premium depending on what the insurance company considers its risk, just as some cars have a higher or lower premium. K-9 Kindergarten and Good Citizenship class discounts, service dog exemptions, people being trained discounts, good dog behavior discounts, etc.
Every dog owner would have to pay something, but many of us already are, with one insurance policy or another.
Some people might lose their dogs from being unable to find realistic and affordable insurance for the breed, but again, they shouldn't have it in the first place. If you can't pay for damages, either with insurance or from your own pocket, you shouldn't be permitted the means to inflict them.