The second ammendment says nothing about any exceptions, and the criminals that matter should be in jail. If they're not, they're unlikely to care about another law anyway. We cannot allow the creation of myriad criminal statutes have the potential to disarm people; we need all the gun owners that we can get.
"something neither group opposes."
And that is the saddest part of all. The second ammendment makes plain that all private guns are outside the purview of government.
Even if they are criminally insane?!
Look, I agree that we need to lock criminals away and take the mentally ill off the streets. The reality is that is not going to happen due to the cost and practicality.
Therefore, keeping illegal aliens, criminals and the mentally dangerous from purchasing firearms is prudent.