Here, we report magnetothermopower measurements on overdoped (Bi,Pb)2(Sr,La)2CuO6+delta (Bi2201) single crystals in magnetic fields up to 35 T. Whereas the temperature dependence of the zero-field Seebeck coefficient S(T) can be captured using Boltzmann transport theory, the field-dependent response S(H) cannot. Instead, the magnetothermopower contains a large additional contribution whose field and temperature dependence is consistent with the presence of short-range superconducting order well above Tc. Combined with earlier Hall and magnetoresistance results, these data imply that the overdoped cuprate strange metal contains two transport sectors with distinct particle-hole symmetry: a conventional particle-hole-asymmetric Fermi-liquid (FL) contribution governing the Hall effect and the zero-field thermopower, and a nearly particle-hole-symmetric sector dominating the anomalous longitudinal magnetotransport. We formulate a phenomenological real-space model in which disconnected FL islands are embedded in a compensated Dirac liquid of phase-incoherent d-wave Bogoliubov quasiparticles. This picture reconciles conventional zero-field transport with anomalous magnetothermopower and magnetoresistance and offers a concrete framework for thinking about strange metallicity in overdoped cuprates.