I will try to give an answer to my question, which is basically an extension of the last paragraph of the question and of the comment of Ryan Thorngren.

I will limit myself to a one-loop study. To this order, the RG flow is the « Ricci flow », $\frac{dg_{ij}}{dt} = - R_{ij}$ where $R_{ij}$ is the Ricci curvature and the variable $t$ is something like - log of the energy scale. I choose this variable only by convenience: the direction of the RG flow from the UV to the IR is the same that the positive direction of the « time » $t$.

This is true for any $X$.

For a metric with positive Ricci curvature, the manifold $X$ is large in the UV, the sigma model is asymptotically free, $X$ shrinks under the RG flow and becomes strongly coupled in the IR. The perturbative sigma model description breaks down when $X$ becomes of size of order $\sqrt{\alpha'}$.

For a metric with negative Ricci curvature, the manifold $X$ is large in the IR, the sigma model is a good description in the IR. It is strongly coupled on the UV, it is not clear if the theory exists in the UV.

For a metric with zero Ricci curvature, we have a fixed point of the RG flow and the sigma model defines a well-defined CFT.

My question can be reformulated as : what happens for a metric which is a small perturbation of a Ricci flat metric, small in particular in the sense of having the same Kähler class. Naively, what happens is not clear because the Ricci curvature of such a metric is neither positive or negative, it is not of fixed sign, the Ricci curvature has fluctuations of both signs around zero. This apparent difficulty was basically the reason for my question. So now the question is: how small fluctuations of the metric evolve under the RG flow? Are there smooth out or are there amplified?

I think that the key point is the remark that $R_{ij}$ is roughly (in correct coordinates and up to non-linear terms) minus the Laplacian of g.To find the Laplacian is not surprizing because the Ricci curvature is by defintion roughly the trace of second derivatives in the metrics. The key point is the minus sign. It means that up to non-linear terms, the RG flow is roughly$\frac{dg_{ij}}{dt} = \Delta g_{ij}$ i.e the heat equation for the metric. This implies that under the RG flow, the fluctuations will be smooth out.

So a small fluctuation of the Ricci flat metric will flow in the IR to the Ricci flat fixed point. In the given Kähler class, the Ricci flat fixed point is the only fixed point and it is an attractive point: all the trajectories converge to this point in the IR.

Toward the UV, the RG flow will have exaclty the inverse behavior: if one tries to go to the UV, the fluctuations will be amplified (as for a heat equation with time inversed). If we begin with a random fluctuation and go to the UV, the total size of $X$ does not change (the Kähler class is unchanged) but the metric on $X$ will fluctuate more and more drastically and apparently chaotically. The perturbative sigma model description will break down when the typical size of the fluctuations will become of the order $\sqrt{\alpha'}$ and it is not clear if there is some definition of the theory in the UV.