im aware of 3 methods for arbitrary fft resynthesis: ifft, additive, or vocoding.
ifft may be possible but is probably about as hard as the original fft. here's one approach using the regular fft to construct the ifft.
As far as additive or vocoding, you create a sine osc or send noise through a bandpass with amplitude equal to the band for each band of the fft. this would be equivalent to having either 512 sine oscs or 512 band passes running simultaneously. so good luck.
However, if the band centers have an integer harmonic relation (i.e. are parts of the overtone series of the lowest band's center) and the lowest band is centered at f, the resynthesis could be done by constantly updating a wavetable with length 1/f*48000 with a weighted sum of the wavetables for each band's sine. the harmonic requirement guarantees that all of the sine waves will complete an integer number of periods per each period of the fundamental, guaranteeing that the end of the table will line up with the start of the table regardless of the weightings, preventing the introduction of unwanted transients. Whether this is any more useful in practice isnt something I can comment on but its an option. Still probably too much going on for the axo since for an f0 of, say, 30hz, this will require 1600*512 multiplications AND additions per wavetable recalculation.