Deconvolution of shifted gaussian function in the frequency range
It is naive, but you didn't tell why you don't use $s(t) = A(t) \cdot \exp(-i \omega_0 t)$, and calculate $A(t) = s(t) \cdot \exp(i \omega_9 t)$...
I named your signal $s(t)$ here.
Other not so naive things.
You mention number of points, so I assume you are discretizing the signal. And your Fourier transform is a finite length DFT. In that case the Fourier transform will have alias, if you sample rate is enough to capture all the information in $A(t)$ then you can simply calculate $S(\omega) = DFT(s(k T_s))$, and the $A(\omega) = S(\omega + \omega_0~ \textrm{mod}~ F_s)$. Where $T_s$ is the sampling period, and $F_s$ is the sampling frequency, thus $s(k T_s)$ is the k-th sample of your signal.