Microwave-dressed polar molecules offer a route to lattice quantum simulators in which the angular form of long-range dipolar interactions, not only their overall strength, can be engineered. We study this setting in a hard-core Bose lattice model on a square optical lattice, with particles interacting through a sign-changing non-axisymmetric dipolar tail (V(r)(x^2-y2)/(x^2+y2)^{5/2}) that is repulsive along one lattice axis and attractive along the other. Using worm-algorithm path-integral quantum Monte Carlo simulations, supported by a hard-core spin mapping and a Gutzwiller soft-mode diagnostic, we find two regimes controlled by (t/V): at larger (t/V) the system remains superfluid but develops a pronounced directional stiffness anisotropy, while at smaller (t/V) it forms a sequence of commensurate stripe solids selected in the ((q,0)) axial family, corresponding to real-space stripes parallel to (y). The leading ordering wave vector remains in this axial family but reorganizes between commensurate filling plateaus, showing that the robust ordered object is a family of stripe states rather than one fixed Bragg peak. Near the closure of the stripe lobe, averaged observables can mimic a narrow supersolid signal; measurement-resolved stripe structure-factor histograms instead reveal first-order switching between superfluid and stripe-solid sectors. NaCs lattice estimates place the relevant (V/t) window within reach of modest effective dressed dipole moments, linking the predicted stripe-family order and its experimental diagnostics to accessible molecular quantum-simulation scales.