I know Denial is knowledgable and capable enough to care for a lynx or a bobcat, but for every one Denial there are probably five joe shmoes off the street who want a tiger because it's cool. Just like with large snakes. Let's say there's a one time license fee of, oh, 125 bucks. That's how much it is to get a driver's license in the province of Ontario, and I'd liken the risk of owning a big cat or a severely venemous snake to the risk of driving a car - you probably don't have to worry if you know what you're doing and you're attentive, but the risk is there. After that, there's a periodical renewal, maybe at no cost, maybe with a negligible processing fee.
I think a system like that would go a long way towards keeping the public and the suits happy, without barring the animals entirely. The catch is. the people who would want to bar private citizens from obtaining permits.