@ Martin - As you say, tearing a few roots that are stuck to the tape is neither here nor there. In fact, I'm even beginning to question the need for any tape at all. Simply cutting a narrow 'window' into the container near its base with a box cutter then covering up again with the black sack should also do the trick. To inspect root growth, I would only need to remove the black sack, which has to be done anyway even for transparent containers. I doubt the moss would 'leak' if the window is narrow enough.
I also have lost leaves from my largest layers, and wondered if they might be dying. But since dormancy was their normal state around that time anyway, I wasn't too concerned. Sure enough, by the following spring, new shoots could be seen pushing thru every time.
My guess is that separation from the parent tree can induce premature dormancy as a survival mechanism, but the new tree always recovers. Luckily for us fig lovers, they seem to be very resilient trees, and their ability to shed leaves probably has a lot to do with that.