I wonder if a kind of whistleblower's reward could help with enforcement? Restaurants have to display their business licenses and food safety certificates publicly. What if hosts also had to display their registration certificates publicly? Guests who can't find the certificate could report the host and receive a reward if the host is found to be violating.
The hard part would be spreading awareness of the reward to all possible guests.
The better solution, although Airbnb has fought this, is to pass regulation forcing hosts to register with the city and forcing Airbnb to validate the host's registration through some means. Airbnb postulates that it's not up to it to figure out if a host is violating the law, but they deal with tax regulations efficiently, and I think this is an extension of that.
The hard part would be spreading awareness of the reward to all possible guests.