A circular situation seems to me, since you acknowledged assumption of rigidity which only then in conjunction with energy redshift forces the force redshift to hold. But admittedly it's logical to do so.
You may have noticed a certain dealing with it last post of mine.
Thanks for putting it professionally and formally that way - but I guess you are basically saying yes.
Sure agree with all these caveats and as extended further in your post, but OP specified the static spacetime SC setting and further I think it quite clear material elastic effects were not under consideration here.
Phew - Peter do you realize we are in agreement on at least two points in all this!?
On that high note I retire for some much needed R and R.