You have quite a few presumptions in that. First, employers don't necessarily need to know if their dancers are licensed. Perhaps it is simply a small misdemeanor + fine with penalties limited to the dancer. In that case, employers don't need to know anything. I haven't looked it up, but there are plenty of reasons why an employer might not care.
Furthermore, the government need not provide any more information than "yes-licensed" or "not-licensed." There certainly does not need to be a directory of dancers, for sure, nor should there be (hello stalker heaven).
One of the major problems for strippers is stalkers. Men get attached, and sometimes it can be become a real problem. You can say that it goes with the territory, but it isn't a problem if the government isn't keeping a publicly-visible directory of dancers, and I say the problem lies there.