It would be nice if the offshore industry would say that the flying medical covers it or vice versa then that would one less expense but since when did logic ever interfere with bureaucracy.
Solar, I was in more or less the same situation a few months ago. My JAA Class 2 medical was about to expire, I needed a new diving medical and I needed another medical check related to a medical procedure from a while ago.
The diving license was easy. Apparently my AME was well used to that and it was only 15 euros or so extra to issue me the bit of paper. The other medical check required some phonecalls between the AME and that other place, to know what exactly was required. As it turned out, it was just a little more detail on the blood work which the AME lab could handle perfectly well. They even gave me a separate invoice for that, to send to the other place, together with the results of the blood work.
So a bit of phoning around saved me at least one medical examination, which otherwise would have required me to take half a day off work.
My advice is to find out *exactly* what is required for your offshore medical, take those bits of paper to the AME and see if they can sign you off for that as well. Ask for a separate invoice and send that to your offshore employer.