As I understand it, the energy gain from a slingshot is derived from the translational energy of the more massive object, and not from the spin.
I have heard that said and GR isn't my thing but I would check that, layman say all manner of dumb things. It could just be you can ignore it for stuff NASA has done because the objects are rotating slowly like Earth. My gut feel is if the object isn't perfectly symmetrical then there must be an effect from the frame drag.
Wiki says frame drag matters at the bottom deals with issue (
https://en.wikipedia.org/wiki/Gravity_assist ) but I would like to see a calculation first.
I know for example the moon orbit and recession is still out fractionally from what is predicted so you need to check these things. Creationist are all over that one and I actually once wondered whether anyone had done the calc.
The slowing down of celestial object's spin is due to tidal effects, not possible with a Born rigid material.
You know my answer .. not worth wasting time to look at ... they can't exist by too many lines of evidence to be wrong. Leaving that to you to worry about.
The spin of neutron stars has been shown to decrease but as yet there is no similar demonstration for black holes.
How would you "conclusively measure" the spin of a black hole if it's inside an event horizon
You have obviously made the choice of a born rigid body in the middle still so I guess we wait for your full theory before I can comment further. I will leave you to it.
