If the flight doc isn't qualified to, or no other docs in your clinic or flight medicine aren't qualified to give FAA medicals, it has been my experience that they know nothing or next to nothing about the topic.
Why spend the money with them when if you are a DSG flying for the airlines you need a class 1 anyways or you get an equivalent class 3 by just having your medical?
Class 2 does almost nothing for people doing what we are doing. You don't even need a class 2 to civilian flight instruct so why bother with spending extra money for it?
I i threw out this path as an option because it is free and the OP wasn't asking about getting a FC1 for the airlines.