No constitutional right is absolute. Not every gun law - in the opinion of SCOTUS (and that's what counts) - constitutes an infringement.
As applied to the subject of background checks, given that Heller makes it clear that felons and severely mentally ill persons may be forbidden to own and possess firearms. Given this fact, at least some background checks are, therefore likely constitutional. (This does not mean they are effective, just that under Heller they are likely constitutional.)
If the legal system would do their job and deal with criminals, 95% of our problems would be gone. Secondly, if a person loses their right to certain constitutional protections their ID should be marked as such just like is done for minors, physically limited, those with convictions of DUI/DWI. Mark their ID, and background checks are done with. It is already illegal for anyone to sell to a disqualified person. If ID's were marked anyone, anywhere, anytime could handle a legal transaction.
Let's quit trying to make this, or support, more complicated than it needs to be. Why won't this happen? Because it would shrink the government employee base, and it would take the sense of power out of those pushing for control of our lives.