Several years ago, I contacted a local FAA FSDO to convert (add) my type ratings on the CAD license onto the FAA ATP. In turn, they contacted OKC (FAA HQ). The short answer is they wouldn't convert the type rating for a part 129 carrier.
Things like this CAN vary FSDO to FSDO. I'd give it a try at a local FSDO and let us know what the answer comes back (and where you went). Not optimistic based on my experience--sorry. Maybe things have changed.