Thank you for the links you have provided, some of which it turns out I have visited before. I have seen Couchoud's work referenced in Claggett and others, but it has seemed to me from the contexts that she has not offered any breakthroughs, merely refinements of translations.
The distinction which you note between the exactitude sought and realized in the hekat calculations as opposed to the fudging apparently allowed in the geometrical solutions is an interesting point. I am also not at ease with the word "sacred", but it would seem that they desired to utilize circle and square relationships in order to arrive at various understandings. I argue elsewhere that a paramount concern for them was in regard to squaring the circle for both circumference and area. I believe that the design of Old Kingdom pyramid interiors is derivable from this point of beginning.
RMP 48, I believe, allows for at least the following inferences:
1) that there was a more than casual interest in what can be termed diagrammatic geometry;
2) that they clearly understood that there was a constant relationship (ie, 64/81) between the area of a circle and the area of the square whose sides were the same length as the diameter of that circle;
3) that they would have extrapolated upon the concept and have drawn the square inscribed within the circle, and the circle circumscribed around the square - forcing a formalizing of the square root of 2 understanding;
4) that it is more than likely that they would have asked the question, "if the area of the inscribed circle is in a constant relationship (ie, 64/81) with the area of the circumscribed square, would not the circumference of the circle be in this same constant relationship (64/81) with the perimeter of the square?"
Since each side of the square is the same length as the circle's diameter, this latter inference of course leads to the circle's circumference being equated with 64/81 times 4 times this diameter - the understanding which i believe is being implemented in MMP 10.
That the equality was more or less correct could easily have been confirmed through empirical measurement. Had they already reached a similar understanding via a C = 11/14 times 4 times this diameter "formula" as a result of empirical implementation using their 28 finger royal cubit system, then they could have confirmed the 64/81 result by comparison of the respective unit fraction chains.
I believe that Problem 48 tells us all of this and more. I continue to look for others who have considered this problem in this way as I am sure that someone must have.