But what about with regards to race? Would it be offensive or unusual to talk about "the Mongolian race" (referring to East Asians in general, not people from Mongolia or ethnic Mongols), beyond any offence generated from talking about race in the first place?
It's outdated, and like most outdated racial terms, it automatically sounds racist by connotation. This has less to do with any specifics of accuracy, and more to do with the fact that the era of its popular usage was shockingly racist as judged by current standards. In general, a person who uses a outdated racial term is assumed (correctly or incorrectly) to have outdated racial ideas.
I don't think there is anything wrong in a grammatical sense, which is technically all this site should deal with.
But really there is more to it than that: the reason you cannot use mongoloid any more is that it relates to a discredited theory about all humans belonging to one of three races, related no doubt to the three sons of Noah. I am not sure why you think 'Mongolian' is a good term to refer to all East Asians, but if that is what your theory holds, it's not specifically derogatory - provided you don't mind people saying "You're Australian? Do you belong to the English race or the Aboriginal one?" If you do object, perhaps you need a better theory.
The terms "Caucasoid," "Mongoloid," and "Negroid" are inevitably tied to the racial theories of the people who coined and used the terms. These theories included skin color as a determinative, and they have been abandoned by scientists and reasonable people. The terms remain in use in forensic anthropology and are restricted to classification of people by bone measurements, particularly craniometry.