Progenitor models for the "luminous" subclass of Fast Blue Optical Transients (LFBOTs; prototype: AT2018cow) are challenged to simultaneously explain all of their observed properties: fast optical rise times days; peak luminosities 10 44 erg s −1 ; low yields 0.1M of 56 Ni; aspherical ejecta with a wide velocity range ( 3000 km s −1 to 0.1 − 0.5c with increasing polar latitude); presence of hydrogen-depleted-but-not-free dense circumstellar material (CSM) on radial scales from ∼ 10 14 cm to ∼ 3×10 16 cm; embedded variable source of non-thermal X-ray/γ−rays, suggestive of a compact object. We show that all of these properties are consistent with the tidal disruption and hyper-accretion of a Wolf-Rayet (WR) star by a black hole (BH) or neutron star (NS) binary companion. In contrast with related previous models, the merger occurs with a long delay following the common envelope (CE) event responsible for birthing the binary, as a result of gradual angular momentum loss to a relic circumbinary disk. Disk-wind outflows from the merger-generated accretion flow generate the 56 Nipoor aspherical ejecta with the requisite velocity range. The optical light curve is powered primarily by reprocessing X-rays from the inner accretion flow/jet, though CSM shock interaction also contributes. Primary CSM sources include WR mass-loss from the earliest stages of the merger ( 10 14 cm) and the relic CE disk and its photoevaporation-driven wind ( 10 16 cm). Longer delayed mergers may instead give rise to supernovae Type Ibn/Icn (depending on the WR evolutionary state), connecting these transient classes with LFBOTs.