Pursuing the historical foundation of the Second Amendment in the 1776 Pennsylvania Constitution, I note that the current version of the PA Uniform Firearms Act contains a definition of a weapon as "a firearm which is not loaded or lacks a clip or other component to render it immediately operable, and components which can readily be assembled into a weapon."
That points me in the further direction of the rest of the US legislation, which has in the past forbidden sales of ammunition to certain classes of individual. Although these have mostly been repealed, the ground never addressed the Constitutional implication you make, nor does the NRA impute it, because it does not exist. Had it existed, those laws would have been unconstitutional, and indeed such limitations as do exist on the sale of ammunition also have the same impact, the Constitution does not extend as far as is being claimed.
Indeed, an even wider lacuna exists, in that only one State, New Jersey, forbids dum-dum bullets, completely forbidden by the Geneva Conventions. If anything, therefore, the entire US legal position on ammunition might be illegal in superior International Law even as it stands.