I contacted my regulator today.....response "we are not sure. We'll get back to you."
I think I'll try my insurance company now...perhaps they have a definition of "operator," or will name the states within which we may fly, or not fly, (ie., those who have applied for derogation, and those who have not) when using FAA licenced crew in our TCO registered aircraft, which may, or not be, operated (depending on the definition) by a "management company" based inside the EU or elsewhere.
I feel like someone has stood me in a barrel and told me to piss in the corner = confused!