  BaNi$_2$As$_2$ is a nonmagnetic analogue of the iron pnictide
superconductors, and in its tetragonal state exhibits an incommensurate
charge-density-wave (IC-CDW) and a sizable elastoresistance. In this work,
phonons in BaNi$_2$As$_2$ associated with the IC-CDW and uniform in-plane
lattice distortions are investigated using high-resolution inelastic X-ray
scattering. The in-plane transverse acoustic phonons reveal no softening at
temperatures where the elastoresistance increases strongly, indicating the
latter to be electronically driven. Systematic phonon measurements reveal the
IC-CDW occurs in two stages upon cooling: underdamped phonons first soften to
zero energy well above the IC-CDW ordering temperature, then the resulting
quasielastic IC-CDW fluctuations gradually slow down and coalesce into the
static IC-CDW order. A possible origin for our observations is the IC-CDW in
tetragonal BaNi$_2$As$_2$ being uniaxial, which provides an additional Ising
degree of freedom favorable for disordered IC-CDW modulations, and accounts for
the elastoresistance through a weak coupling to the lattice.
