  Metallic LiOsO$_3$ undergoes a continuous ferroelectric-like structural phase
transition below $T_c$ = 140 K to realize a polar metal. To understand the
microscopic interactions that drive this transition, we study its critical
behavior above $T_c$ via electromechanical coupling - distortions of the
lattice induced by short-range dipole-dipole correlations arising from Li
off-center displacements. By mapping the full angular distribution of second
harmonic electric-quadrupole radiation from LiOsO$_3$ and performing a
simplified hyper-polarizable bond model analysis, we uncover subtle
symmetry-preserving lattice distortions over a broad temperature range
extending from $T_c$ up to around 230 K, characterized by non-uniform changes
in the short and long Li-O bond lengths. Such an extended region of critical
fluctuations may explain anomalous features reported in specific heat and Raman
scattering data, and suggests the presence of competing interactions that are
not accounted for in existing theoretical treatments. More broadly, our results
showcase how electromechanical effects serve as a probe of critical behavior
near inversion symmetry breaking transitions in metals.
