Sources of quadrupole radiation from black hole

During ringdown from BH merger, the event horizon is (obviously) very far from equilibrium. This is a statement about mass distribution in spacetime.

Suppose I fly a rocket across the event horizon (of a quiescient BH). On the one hand, it seems plausible to argue that 100% of the resulting gravitational radiation emitted as a result can be described by the BH + rocket mass distribution before the rocket crosses the event horizon (EH). On the other hand, the Cauchy horizon does not coincide with the EH: so, if, after crossing the EH, if the rocket made a sudden change of direction, could this ever affect the shape of the EH? It seems like the answer is "no", because any radiation arsing from rocket maneuvers cannot travel fast enough to reach the EH from the inside. Right?

By reductio ad absurdum, that would imply that any sudden rocket accelerations, made immediately before crossing the EH, become exponentially(?) damped with regard to the resulting GR radiation. I guess. Is there a theorem that states this? Is this a lemma of the no-hair theorem?

Yes, this question is kind of garbled, since I’m conflating the shape of the EH with far-field gravitational radiation. It’s just that I don’t recall ever reading any clear statement about this; did I simply sleepwalk through that part?

(The obvious application is "can a rocket transmit a message to the outside world, by sharply accelerating just after it crosses the EH?" The orthodox answer is obviously "no", but the details of this is not readily apparent (to me))