I stumbled upon this old question and I thought maybe I could have something to say.

As it happens to have done a lot of work in holographic renormalization it turns out it is indeed possible to do holography and thus holographic renormalization for $AdS_{p+2}/\text{CFT}_{p+1}$ where $p+1$ is the dimension of the CFT using a generalized form of the Fefferman-Graham expansions. In specific holography in the methods I will discuss is possible to be performed because the bulk admits a **conformally** **asymptotically $AdS$ geometry **for $p =1,2,4$. For $p=5$ one gets $E^{5,1} \times \mathbb{R} \times S^3$ which is non-$AdS$ background it involves a linear dilaton and the qualitative picture of holography for both $D5$-branes and $NS5$-branes is different. Now, there are two methods to proceed. One is to take the action of the low energy SUGRA which involves a conformal factor and go to a string like frame perform holography there and then return to the original frame of the action. The other way is something called "generalized dilatation operator method". There one begins with the usual ADM foliation of the manifold, using the action in the original Einstein frame directly (thus more efficient already) in the radial $AdS$-like direction and finds asymptotic radial expansions the bulk fields near the boundary. The counterterm action that removes the divergences due to the infinite volume is nothing else than Hamilton's principal function $\mathcal{S}$. Then is not very complicated to find the holographic dictionary. Note that both methods fail to work for $p=6,7,8,9$. At least for the second methid, which I am familiar with, the asymptotic radial expansion fail to be correct. I am not sure on the progress for the cases of $p=6,7,8,9$.

By the way, let me mention that this process of the generalized dilatation operator method, in some sense, it is useful in the context of formulating the Hamiltonian problem in cases where the initial/final values of the fields asymptotically approach zero or infinity in problems that do not necessarily admit a holographic dual. It is a very useful concept indeed.