First off, you do not gain energy by changing direction.
NOONE SAID YOU DID.
What was said is you create an infinite force !!!!!!!!
Force = change in momentum / change in timeAs change in time goes to zero force goes to infinity.
I will ignore the light bit it is irrelevant.
Second, their is no radiation from a rotating black hole. Kerr is the asymptotic solution and contains no radiation.
WOW maybe we need to stay in mathematics you actual make sense in that domain.
You are correct if and only if the black hole is absolutely perfectly symmetrical.
Now try the solution on a feeding spinning black hole on even just one object and tell me what happens.
I will give you a statement of the solution, that gravitational waves are radiated by objects whose motion involves acceleration where that the motion is not perfectly spherically symmetric.
Now use your mathematics to prove or falsify that statement.