Requiring a permit is merely a time, place, and manner restriction, and may not be exercised in an arbitrary and capricious manner. Content may not be regulated. (This is why the Nazis were allowed to march in Skokie). Because free speech is a fundamental right, the regulation must be narrowly tailored to prevent the type of harm that may come from the speech-related activity, and the regulation must be no broader than necessary. This is the "strict scrutiny" standard that is used by Courts to evaluate whether or not a restriction on a fundamental right is overbroad.
Using this standard, I can't think of any harm that would be prevented by a national gun registry, as a registry will prevent no one from misusing their gun.
Thoughts?
Pretty good approach - does the registry have to show utility? Unfortunately, most government programs would fail this test as well, but haven't been ruled unconstitutional.