You have brought up the concept of licensing, and the seperate concept of background checks.
Licensing is a precondition to the exercise of the right not inherent to same. This is an infringement. This differes from the precondition of voter registration because voter registration determines, most imnportantly, where your vote is to be cast, which IS an inherent part of the right to vote -- and so, is NOT an infringement.
A secondary effect of voter registration is to determine the identity of the voter. This determines name, age, residency, etc -- all inherent parts of the right to vote, and therefore, not an infringement. This is the same as presenting an ID when buying a gun.
Background checks create a form of prior restraint. This is an infringement. Prior restraint is the condition of the government preventing you from exercising your right on the basis that you MIGHT commit a crime if you were to be free to act -- your right is withheld 'just in case'. This is exactly what a background check does, 'just in case' you are a felon, etc.
No one has ever argued that voter registration is prior restraint, becase, of its face, it is not.