Periodic driving offers a powerful tool to engineer topological phases and even induce phase transitions that are inaccessible in static settings. In this work, we demonstrate that a suitable driving protocol applied to the Benalcazar-Bernevig-Hughes (BBH) model, a canonical quadrupolar insulator with a $\pi$-flux-induced projective $𝒫𝒯$-symmetry, gives rise to a topological phase in which the dispersive first-order edge states and localized higher-order corner modes are shown to coexist at distinct quasienergies. Extending the scenario to non-reciprocal hopping induced non-Hermiticity, we uncover a ${\mathbb{Z}}_2$-like skin effect characterized by a sharp transition from a unipolar to a bipolar eigenstate localized phases. This phenomenon, which is established as a prerogative for spinful systems, emerges here purely from the interplay between the embedded gauge structure and Floquet-renormalized symmetry constraints, without invoking physical spin degrees of freedom. The broken bulk-boundary correspondence in this driven non-Hermitian setting, can be restored via computing the two-dimensional generalized Brillouin zone (GBZ), which we construct through a symmetry-reduced mapping onto an effective one-dimensional problem. The resulting non-Bloch invariants faithfully capture both the higher-order topology and the unipolar to bipolar transition. These findings reveal periodic driving as a versatile and controllable handle for sculpting the interplay of higher-order topology, symmetry transmutation, and non-Hermitian skin physics in a single unified platform.