The thing to note here is that the theory of epicycles was never actually a thing; contrary to popular belief, geocentric astronomy did not fix its problems by

The thing to note here is that the theory of epicycles was never actually a thing; contrary to popular belief, geocentric astronomy did not fix its problems by adding additional layers of epicycles. It had one layer of epicycles, that was it; it did have additional complications, in the form of the eccentric and the equant, but these additional complications were not additional layers of epicycles that could be stacked arbitrarily. (Copernicus may have used a second layer of epicycles in his heliocentric system, which eliminated the equant (but not, I believe, the eccentric)? Admittedly two does suggest the possibility of generalizing in a way that one doesn't, but I don't think that generalization was ever performed at the time before Kepler made the whole thing irrelevant.) Had it been used, it basically would have been Fourier series, yeah! I don't know whether that counts as "perturbation theory" proper. But it's worth noting that the real development didn't happen that way, and the further corrections that were actually used were the eccentric and the equant, not further epicycles!