I found the inclusion of licenses in SR5 to be a bit baffling, given the setting.
Remember that bit about extra-territoriality that Megacorps have (which is a core to the setting)? That means that when you're on SK grounds, you're working under their laws. In all likelihood, your average runner doesn't really care about SK's laws, unless their job actually requires them to work within their target's jurisdiction as a legally-represented individual.
Similarly, when you're in Denver, what permit do you use? Do the various states represented in Denver have mutual agreements to recognize a license issued by Aztlan in UCAS territory?
I think what makes sense depends on how much paperwork and tracking you like to foist on your players. If you're really into the number-crunching and keeping track of everything like that, then that means you have to read the rules as given and apply them in the extreme literal. Given that, and that there are no licensing rules for melee weapons vs firearms, you can just assume you have to have a license for Restricted weapons, no matter the type.
If you're just throwing it in as flavor, or as a key part of a particular run, have the Johnson provide IDs, or require the PCs to get licenses for that run. Record them, use them, toss them into the pile of "stuff my guy's picked up in the past." Note that if you get a burner-license, after a job, depending on how stealthy the run was, it may no longer be usable, and using the license in many separate jobs may help someone else track the runner down.
Another thing you could do is, during char-gen, if PCs want to buy a set number of licenses to be used in the future, have them spend the cred, note down "x blank licenses" on their sheet, and if they run into a situation where they need to get a license fast, convert one of them into a specific license at that time.
I kind of like the idea of applying a FATE style player-bump, here... if they're on a run, and they realize after the fact that they should have gotten a license to get through a particular point, have them spend an Edge to say, "Oh yeah, of course I put that in my other pocket!" It shouldn't be used to defuse critical parts of your run (you might want to add in an opposed Con or Etiquette skill check if their license is a fake, to keep them on their toes), but if the goal is to add flavor and keep the game moving, this might be one way to accomplish that.