A background check is an infringement, period.
Let's say a ex-felon buys a gun. There's only one circumstance when that is a problem, if he uses weapon in a crime.
If he does not a commit a crime with the gun, then why should anyone care? Who is harmed?
Maybe he has legitimate concerns about the safety of his family from his prior associates. Maybe he's a small business owner with cash on the premises and he works late into the night.
He has the same rights as anyone else...or hasn't he paid his debt? If his debt wasn't paid after he was released from prison and completed any probationary term the law required...why then was he released?
But, boy, do those background check empower the government to snoop in to people's lives, or don't they? They sure do.
Nope, the ex-felon should be just as able to purchase and carry a firearm as any other citizen.
But if the ex-felon is convicted of USING the weapon in a crime, or merely having the weapon in his possession when he comits a crime, he should be punished most harshly. Clearly the effort to reform him failed, so punish him until he's really old and worn out, as an example to others.