It is known that a given smooth del Pezzo surface or Fano threefold X
admits a choice of log Calabi-Yau compactified mirror toric Landau-Ginzburg
model (with respect to certain fixed Kahler classes and Gorenstein toric
degenerations). Here we consider the problem of constructing a corresponding
map Theta from a domain in the complexified Kahler cone of X to a
well-defined, separated moduli space M of polarised manifolds
endowed with a canonical metric. We prove a complete result for del Pezzos and
a partial result for some special Fano threefolds. The construction uses some
fundamental results in the theory of constant scalar curvature K\"ahler
metrics. As a consequence M parametrises K-stable manifolds and
the domain of Theta is endowed with the pullback of a Weil-Petersson form.
