The marine insurance regime is quite complex. In a situation like that where there's a liability issue, the “protection and indemnity clubs”, as they are called, would come into play, and it would be the club that the ship used as its indemnity provider.
My experience is that, yes, they do become involved, but as for how involved they would be in determining how the response is carried out, I think the marine insurance industry itself would be the best one to answer.