Physics of the multiferroic interfaces is currently understood mostly within a phenomenological framework including screening of the polarization field and depolarizing charges. Largely unexplored still remains the band dependence of the interfacial charge modulation, as well as the associated changes of the electron-phonon interaction, coupling the charge and lattice degrees of freedom. Here, multiferroic heterostructures of the colossal-magnetoresistance manganite La1-xSrxMnO3 buried under ferroelectric BaTiO3 and PbZrxTi1-xO3 are explored using soft-X-ray angle-resolved photoemission. The experimental band dispersions from the buried La1-xSrxMnO3 identify coexisting two-dimensional hole and three-dimensional electron charge carriers. The ferroelectric polarization modulates their charge density, changing the band filling and orbital occupation in the interfacial region. Furthermore, these changes in the carrier density affect the coupling of the 2D holes and 3D electrons with the lattice which forms large Froelich polarons inherently reducing mobility of the charge carriers. We find that the fast dynamic response of electrons makes them much more efficient in screening of the electron-lattice interaction compared to the holes. Our k-resolved results on the orbital occupancy, band filling and electron-lattice interaction in multiferroic oxide heterostructures modulated by the ferroelectric polarization disclose most fundamental physics of these systems needed for further progress of beyond-CMOS ferro-functional electronics.