During the inflationary period the infant universe had to expand very much faster than the current speed of light. However, the universe was so tiny at the time that at the end of inflation it was only a few centimetres across (So I understand from reading, I don't actually have the ruler with the markings.)
The infant universe consisted, too, of nothing but radiation; so the workaround seems to be that it's OK for spacetime to expand faster than the speed of light. Of course, there was no information in the infant universe, so the rule that information cannot travel faster than light, at least, wasn't being violated. But if it was expanding at the current variable speed of light, different parts would still not be in communication with one another, so anisotropy would still be possible.
I don't actually see the problem if the speed of light is related to, say, the logarithmic scale of the universe, since this has effectively been more or less constant for a very long time.
But IANAP, so perhaps someone who is will enlighten me.