Actually we do. Our Constitutional rights are not limited to what is specifically articulated in the Bill of Rights. Indeed, some of the founders argued against a Bill of Rights for fear that some with tyrannical predispositions would come to understand that the only individual liberties would be those that were written down, when such was not their intent. I doubt they really ever contemplated a time when government would want to have a role, or the people would want to let the government have a role in telling them what kind of animal(s) they could own, but I'm sure a lot of the nanny-stater supporters are glad to subvert their intent in this area as in most others.
Specifically, if a right is not explicitly withheld or reserved by the Constitution, or further modified as delegated to the individual states, it's a Constitutional Right. Certainly the states can enact bans against them, but that is up to the individual states. Likewise, the interstate commerce clause has become so distorted and misinterpreted that the Federal government can now see fit to regulate anything it pretty much pleases, so we're now at the point where Constitutionality has very little to do with how we actually operate...which has allowed the collective wet dream for those who would tell the rest of us how best to live our lives and what kind of dogs we might own, how much salt we can put in our foods, what kind of helmet you have to wear when peddling your bike, etc.
Well said.