That’s a really interesting point. I wonder if there’s any proper theoretical work done on this possibility.
My immediate, naive instinct is that by crossing this limit, the frame dragging effect, would be extremely powerful, to the point where it might increase the radius where Hawking radiation is formed/emitted and increase the rate of Hawking radiation to avoid passing the limit. A sort of self limiting process to prevent breaking the speed limit of c.
But in the time it took to write this out, I remembered angular momentum and realised that notwithstanding the additional angular momentum of the infalling mass, the conservation of angular momentum would just make the event horizon slower as it expands. Which makes the superluminal event horizon unlikely in my mind.
It was response to deleted comment about "sticking shell" inside of a black hole, to illustrate that such shell will have superluminal speed, thus it cannot exist, because any matter will decay.