Most people would consider global symmetries different from isometries. On the global symmetry side, it's possible to give a short answer which is that gravity doesn't have (continuous) global symmetries. For any localized excitation, there is a dynamical process which puts it behind the horizon of a black hole, thereby preventing us from associating it with a conserved charge. I should add that it's possible to have global symmetries in the bulk if you interpret AdS/CFT more generally to include non-gravitational QFTs in AdS which have boundary dynamics described by a nonlocal CFT. This setup is essentially equivalent to defect conformal field theory in which there are two possible fates of a global symmetry. If it is preserved by the defect then nothing happens to it. Otherwise, it results in a tilt operator on the defect which is exactly marginal and obeys the Ward identity
\begin{align}
\int d^{d-1}x_{\parallel} \left < t(x) \phi_{a_1}(x_1) \dots \phi_{a_n}(x_n) \right > = \sum_{i = 1}^n (R)_{a_i}^{\;\;b_i} \left < \dots \phi_{b_i}(x_i) \dots \right >.
\end{align}
It is similar to the displacement operator which has protected dimension $d + 1$ and obeys
\begin{align}
\int d^{d-1}x_{\parallel} \left < D(x) \phi(x_1) \dots \phi(x_n) \right > = \sum_{i = 1}^n \frac{\partial}{\partial x^i_{\perp}} \left < \phi(x_1) \dots \phi(x_n) \right >
\end{align}
along with some recently derived sum rules.
But then your example gets at something very open ended which is how do we understand the isomorphism of Hilbert spaces in AdS/CFT from the point of view of isometries? To start, we know that the ground state is empty AdS which preserves the maximal number of isometries. These correspond to conformal symmetries on the boundary which preserve the vacuum of the CFT. We can then go through the local operators which furnish non-trivial representations of this symmetry. By the state-operator map, these create less symmetric deviations away from pure AdS. A scalar primary will give $\phi(0) \left | 0 \right >$ which preserves the $SO(d)$ rotations $M_{\mu\nu}$ but it will have a non-trivial scaling dimension for instance. If you want something that preserves $SO(d - 1)$ spatial rotations but not boosts, we can look at the action
\begin{align}
[M_{\mu\nu}, v^\rho \mathcal{O}_\rho] = v_\mu \mathcal{O}_\nu - v_\nu \mathcal{O}_\mu
\end{align}
on a vector primary contracted with some auxiliary polarization. Clearly, a spatial $M_{xy}$ will give a commutator of zero as long as the only non-zero component of $v^\rho$ is $v^t$. But there are infinitely many other things we could've done to achieve this result.
Coming to black holes now, for some purposes it is possible to avoid the question of what the particular operators are that create this state. At temperatures where black holes dominate the canonical ensemble (as opposed to say a gas of gravitons), you can think of black holes as being dual to a thermal state
\begin{align}
\rho = Z^{-1} \sum_n e^{-\beta E_n} \left | n \right > \left < n \right | \to Z^{-1} \sum_n e^{-\beta u^\mu P_\mu} \left | n \right > \left < n \right |.
\end{align}
The details of the theory go into determining an equation of state. But once this is known, you can write down the coefficients in the stress tensor
\begin{align}
T^{\mu\nu} = p \eta^{\mu\nu} + (\epsilon + p) u^\mu u^\nu
\end{align}
which contains a fluid velocity $u^\mu$ breaking boosts. If you write the metric of an AdS-Schwarzschild black hole in Fefferman-Graham form, you can read off this stress tensor (along with a current if it is charged). Promoting the parameters of a black hole to local fields subject to Einstein's equation will then lead to velocity fields in $T^{\mu\nu}$ that obey the equations of hydrodynamics, an interesting result known as the fluid-gravity correspondence.
A different direction to go in is to consider the actual black hole microstates and this is where the protected operators you refer to become important. We should recall from the holographic dictionary that semi-classical gravity is only valid when the effective string coupling $g_s N$ is large. On the other hand, we will understand the CFT side best when $g_s N$ is small. We would therefore like to specialize to states whose energies are protected from coupling corrections. The thing that can protect them is supersymmetry and therefore they will be dual to microstates of BPS black holes. I.e. the Reissner-Nordstrom black holes which have the minimum possible mass to prevent a naked singularity given their other charges. The first question to answer about these microstates is whether there are the right number of them. In some sense, this was answered before AdS/CFT by Strominger and Vafa. They computed the area of some BPS black holes in the supergravity limit of type IIA string theory and then considered a weakly coupled description of these black holes given by $N$ branes that had one direction (time) of infinite extent. But then they T-dualized to type IIB to get two extended directions which gave them access to 2d CFT methods for counting states. They ended up with
\begin{align}
\log \rho = \frac{A}{4G} + O(1/N)
\end{align}
verifying Bekenstein-Hawking. There were other dimensions in these branes but they were compactified on a manifold which was chosen to preserve supersymmetry. Without this, all states would have energies depending on $g_s N$ and it would be too hard to tell which ones correspond to which black hole.
The Strominger-Vafa argument has some advantages like working in flat space. But it also has the disadvantage that it is mostly about counting. To gain a more refined characterization of black hole microstates, it helps to look at AdS backgrounds where there is a known Lagrangian for the dual CFT. If you look at BPS operators in $SU(N)$ maximally supersymmetric Yang-Mills theory at weak coupling, most of them will turn out to have more in common with gravitons than black holes. But a counter-example was recently found by Chang and Lin. Their search was computationally intensive already for $N = 2$ but it revealed the important lesson that BPS black holes are fortuitous. I.e. the fact that they are BPS relies on trace relations which stop holding at larger values of $N$. Raising $N$ should eventually lead to more and more fortuitous operators with dimension of order $N^2$. But we will need to know a lot of them before we can see the isometries of black holes emerging from the whole set as a statistical property.
