They have told you that you can not register the vehicle until it is operational. You ,yourself, acknowledge that you must be able to pass inspection in your question
but I live in Texas, specifically a county where registration depends on a safety and emissions check.
You work on the car and get it to the point where it passes their inspection.
At least in NC, the law is unclear on the issue. However, it is my (and thanks to Brian R's comment, not everyone's) experience that such signs, flashing or with posted times, are posted every few feet within the zone.
Thus, even if the first sign you pass was not flashing, if subsequent ones begin flashing, you should reduce speed.
It may be a good idea ...