As the title says, does CFT in AdS/CFT live in flat spacetime, or is it only approximately flat?
The CFT in AdS/CFT lives on the asymptotic boundary of the AdS space. The asymptotic boundary is a conformal manifold associated to any "visibility manifold". See this paper of Eberlein and O'Neil (1973), section 1. They define points on the asymptotic boundary as equivalence classes of geodesic rays where geodesic rays are considered equivalent if their distance is asymptotically bounded from above.
One can thus think about the tangent space of the asymptotic boundary to be the normal directions to a geodesic along which nearby geodesics can diverge. This lets one define a conformal structure. There is clearly no natural Riemannian structure on the boundary (and reparametrizations of the geodesics act like Weyl transformations) so if the boundary theory is to be defined in terms of the bulk, it has to be conformally invariant.
However this classical picture breaks slightly when quantizing, and in the end we do have to choose a metric to regulate our theory (this is the case with almost all CFTs). Then we get some mild dependence on the chosen metric through the Weyl anomaly.