Magnetic impurities in real materials can mask the intrinsic spin-dependent properties of hosts. They interact indirectly through the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism, which limits the use of isolated impurity spins in applications such as qubits and spintronics. By suppressing the RKKY interaction, an effective control scheme should therefore enable access to the host's unperturbed behavior while simultaneously isolating the impurity spins for functional use. Although beyond static approaches, single-color laser driving can suppress the RKKY interaction, it typically requires strong fields that may be impractical or destabilizing. To overcome these limitations, we show that two-color laser driving provides efficient and tunable control over all components of the RKKY interaction using two weak laser fields. Focusing on two-dimensional Rashba altermagnets, we show that interference between one- and two-photon processes produces altermagnet-specific Floquet corrections. These include additional AC Stark shifts, magnetizations, spin-orbit renormalization, and emergent in-plane Zeeman fields, which are absent under single-color driving and in non-altermagnetic systems. Notably, two-color driving induces a finite z-component of the Dzyaloshinskii-Moriya (DM) interaction, stabilizing in-plane chiral magnetism and related textures in Rashba altermagnets. These effects enable tunable, near-complete on-off switching of the Heisenberg, Ising, and DM interactions through a Lifshitz-like modulation of the Fermi surface. We also show that the tuning process is highly sensitive to the chirality of both beams. We further map out phase diagrams for ferromagnetic and antiferromagnetic alignment of impurities with clockwise and counterclockwise canting as a function of Rashba coupling and altermagnetic order. Finally, we discuss candidate material platforms and experimental feasibility.