Do I believe a felon who's served his or her time should be permitted to arm themselves? Absolutely. They did their time, and should be considered citizens in the fullest again. If the crime was so serious as to have their rights permanently suppressed then they shouldn't have been let out in the first place.
Do I believe a crime committed with a weapon should bear more responsibility than a crime without a weapon? Absolutely not. Crimes are crimes regardless of the implements used. Punish actions, not objects.
These are examples of moral issues, though. These are not strictly constitutional issues (though the first is debatable), just as you identified. Further, these (punitive statutes) are issues that per the 10th Amendment are relegated to the States to decide. I will not delve into the subject of state statutes, because I frankly don't know other states' statues very well. I can argue Florida law, sure, but that's because I know the Florida Constitution fairly well. I can't say the same about, say, Michigan and Missouri. The only laws we all have in common are those on the national level, hence, those are the only laws I can fairly argue/discuss.
There is nothing, not one thing in our country's Constitution that bars Congress from ensuring the militia, We The People, are well-regulated. Well-regulated meant disciplined. Trained. Ready. In fact, that's exactly what the first Article of the Constitution states as one of Congress's duties. The right to keep and bear arms, as outlined by the Constitution, was not a prohibition on training, and the literature of the Founders made pretty damn clear that they wanted an armed, trained, and ready citizenry. That right to keep and bear arms we as humans are endowed with is recognized by the Second Amendment as coming with the condition that we, as armed Americans, are always trained and always ready to protect that which needs protection.
So I say again, mandatory training and requalification, provided that they were not prohibitive and were offered freely to U.S. citizens, is an example of both Congress fulfilling its duties and of what I suppose could be called "reasonable" (read: constitutional) gun control. Cost should not be an issue; since it's required, it's the government's job to ensure it's paid for. Those states with more intensive requirements (there will be some) would pay the difference.