In large N strong coupling limit of AdS/CFT, We know that not every CFT state has a bulk dual described by a classical metric. What about the converse? Does any asymptotically AdS classical bulk metric have a dual CFT state?
Given a state in the CFT, one can either causally reconstruct a portion of the bulk using HKLL reconstruction, or obtain the Einstein equations from the CFT using RT formula. In both cases we have a bulk description. So the first statement in your question is not correct.
The basic premise of AdS-CFT is that upon correct matching of symmetries, spectrum and operators in the theory according to the holographic dictionary, there is an equivalence between observables computed in both theories, say correlation functions. This follows from the fact that the partition function on both sides upon using the dictionary match, and by its very nature this duality is at a quantum level. You can see that the theory on the AdS side cannot be just classical gravity, as is raised in your question.
There are deeper questions however on whether the CFT potentially contains all possible bulk states in the full quantum gravity description, i.e. whether all states in the AdS side map to those on the CFT side (I think this was probably what you wanted to ask). A claimed aspect of this question in the language of AMPSS (note that this is the second paper by the firewall people, the first one is just plain wrong) is that the states generated by the action of operators behind the horizon of a black hole are not present in the CFT. However it was shown by PR (also see preceding papers) that there are state-dependent maps from the bulk to the CFT which can describe the behind horizon operators (which I consider is most likely correct).
Again for black holes, there can be bags-of-gold paradox that there can be too many excitations in the AdS bulk. So again there is a question on whether all these excitations are actually present in the CFT description or not (I personally think they are actually present in the CFT description, one does need to cleverly describe them though).