Attempting to buy a gun when you have committed a crime that puts you on the prohibited list is a federal felony. So participating in a background check is self-incrimination, either by admitting you are committing the crime or by lying and thus committing a crime. So courts have already ruled that criminals don't have to engage in background checks.
And they don't: criminals get their guns from sources where they don't have to do a background check, like stealing them, buying them from other criminals, or "borrowing" them from family.
Buying a handgun is NOT a matter of public record -- it can't be, by law. In fact, when a newspaper managed to get its hand on a list of people who had bought handguns, do you know what the result was? An increase in burglary at the addresses of the people the paper published! Making the ownership of guns a matter of public record is an invitation to criminals, making it easier for criminals to get guns.
Verifying annually that you own the gun means the government can come round up the guns. That's been the result historically every time there's been such a law, or any other form of registration. Gun registration is an invitation to a police state, and we're building that too fast already. Even when it wasn't government that held the registration, governments took the information anyway and used it to take people's guns and establish a police state.
Besides which, there's no authorization of such a law in the Constitution. There is authorization for laws which could have an effect, e.g. safe storage of any firearm not in use; that could be done under Article I Section 8 -- for that matter, a law penalizing negligent storage that allowed a criminal to easily take your gun(s) might work (definitely so if a tax credit was given for the cost of safe storage, an expense which would arguably come under Congress' authority to provide for the arming of the militia, on the argument that making sure the militia's arms stay in the possession of their owners is part of arming them, since the loss of their guns would mean they got disarmed).