It seems I did a poor job of explaining what I feel is the issue. I will try again.
Consider a "normal" approach, for example one that is straight-in aligned to the runway. You need to circle to another runway. As you come down final, you begin that circle (i.e. no longer following the approach procedure) once you get the airport in sight, and are within the circling radius for your category. That's the "earliest" you can begin it. But once you get to the MAP, you either need to be circling OR executing the missed approach - there isn't really any other choice. Notice that "circling" doesn't necessarily mean "turning" - in a circling maneuver there are times you are flying straight too. So once you get to the MAP, if you aren't going missed, you are by definition circling - even if you're not turning yet. I call this the "latest" point the circling maneuver begins - at the MAP. You can choose to start it earlier, but if you are staying visual and not doing the missed approach, then once you get past the MAP, you are de facto in a circling maneuver.
On the example procedure, though, the final course comes through the circling restriction. And the MAP itself is within the circling restriction. Once you get to the MAP, if you don't go missed, then you are executing a circling maneuver, even if that "maneuver" is "fly straight". Therefore, you are circling within the restriction, which is specifically prohibited.
Furthermore, if you look at the circling restriction example from the 8260.3E, you see that the circling restriction area is NOT evaluated for obstacles, and that this circling restriction starts right at the centerline of the runway. In the figure 2-7-2, those 584 and 603 MSL towers are not used to calculate the circling MDA - nothing in the light orange part is.
View attachment 91242
Now, in the case of Bethel, AK, I'm not really too concerned about the obstacle situation, because the evaluation for the final approach segment pretty much covers anywhere you're reasonably likely to go. And it's only about 1/3 of a nm that you have fly to get into the approved circling area. But that may or may not be true at another airport and procedure with different geometry.
I will also say, that of the 6 examples provided in FAAO 8260.3E about circling restrictions, they all include several hypothetical flight tracks, and none of them come through the restriction. Not that that sets policy, but I did think it was interesting.
I am trying to find another example where the final approach course comes through a circling restriction, but it's a tough search, especially since most desirable would be a MAP that is even further into the restriction that at BET. EEO has a couple, but those MAPs appear to put you basically right over the runway centerline. Same at 20V.