Felons can’t have guns for the same reason that states can decide that you must carry openly or must conceal your weapon — it’s a form of regulation of the unorganized militia.
The right to keep and bear arms can be harmonized with the legitimate concerns of gun-control advocates (as distinct from the desires of statists who want only agents of the government to be armed) if one takes seriously what the Founders understood by a “well-regulated militia” (a well-armed citizenry ready and able to voluntarily, without state control, bear arms in defense of self, family, community and country):
Felons lose their right to be part of the unorganized militia by virtue of their conviction, a manual of arms for the unorganized militia can be promulgated (of which “you must carry openly” or “you must hide your firearm when carrying but not using it” are versions in miniature), and so forth.
No need to harmonize anything with the Empty Set.