Here is the main area where the logic of using prior restraint outside of free speech bothers me. There is not a single condition that prohibits an individual from free speech. There are specific types of speech that are restricted. If an individual were to violate them, he/she may be subject to a penalty, but the basic right to free speech is not removed. Not so with 2nd Amendment rights. Because of the deadly nature of firearms, it became necessary to restrict this right to those who have committed certain crimes in the past. This is typically designated as Class C felonies. Should this condition exist, the individual does not have this right.
The argument that background checks are a form of prior restraint rests on the idea that it is intended to prevent the illegal practice of the act preemptively, by censoring a violation of restricted form of speech before they happen. The intent of them may very well be to reduce violent crime, but the action itself is checking for eligibility, to see if the purchaser has committed a crime that has prohibited him from his 2nd Amendment rights.
As mentioned before, there's nothing here that changes anything.
However you want to word it, the basic function of the background check is to restrain the exericse of the right prior to the exercise of same to determine if said exercise will break the law; if it does, then the exercise is denied, if not, then it it is allowed to continue.
The important part of this is "...to determine if said exercise will break the law..." -- it doesnt matter what law is broken, and it doesnt matter the basis for that law, the issue is the breaking of the law itself. As such, your differential argument as to felons having their right removed is meaningless as it does not matter how the law in question came to be or what effect it has, only that the law is/is not broken by the exercise in question.