The Nazis suggested the name in WW2
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Unhistorical.
The term has been used by most peoples of the area to refer to their country since at least the time of the Sassanids, whose Kings referred to themselves as the “Kings of Aryan and non-Aryan.” With Aryan (Iran) referring to the plateau and mountain country we now call Persia/Iran and “non-Aryan” referring to their conquests outside this area.
You are correct that the term was adopted and exploited (and misused) by the Nazis, but it goes back long before then. Thousands of years in India. They did the same thing with the swastika, BTW, which had a long and generally honorable history before they polluted it.
The Shah requested foreign countries use the term Iran in 1935, not during WWII. In 1935 the Germans were not yet exactly dominant in world affairs.
I’m afraid this is another example of trying to make everything done by other cultures a reaction to our own.