The discovery of integer and fractional Chern insulators in rhombohedral graphene, following their discovery in the isolated topological flat band in twisted MoTe₂, motivates a search for the conditions under which an electronic band can host a topological correlated state. In this Letter we introduce a ``Berry basin" model which captures the essential features of rhombohedral graphene: a dispersive band with a flat bottom, momentum-space winding of a ``layer" degree of freedom, and Coulomb interactions, while also including periodic potentials, enabling controlled studies of the interplay of quantum geometry, superlattice modulation, and electron-electron interactions in producing correlated topological states beyond the isolated topological band paradigm. While integer Chern insulators emerge at strong coupling without a superlattice potential, stabilizing the fractional state requires nonzero potential for all interaction strengths we access. We provide insight into the greater robustness of integer and fractional Chern insulators in rhombohedral graphene when electrons are polarized away from the moiré interface. Our results establish a new class of systems that host robust fractional Chern insulators and provide a controlled numerical platform to test competing theories of interacting topological phases.