The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between impurity spins is calculated for bilayer graphene in the presence of a layer symmetry-breaking external electric field. We find that for intercalated impurities (i.e., impurity atoms between the two constituent layers of the bilayer) the interaction is extraordinarily sensitive to such a field. In particular, (i) the form of the RKKY interaction may be tuned between oscillatory, ferromagnetic, and antiferromagnetic simply by varying the external field, and (ii) the strength of the RKKY interaction may be increased by an order of magnitude by application of an external field. This sensitivity arises directly from the "Mexican hat" form that the low-energy spectrum takes in an applied field. These finding suggest that heterostructures of intercalated magnetic atoms in bilayer graphene may represent a possible system for electrical control over magnetic structure.