The integration of InAs/GaAs quantum dots into nanophotonic cavities has led to impressive demonstrations of cavity quantum electrodynamics. However, these demonstrations are primarily based on two-level excitonic systems. Efforts to couple long-lived quantum dot electron spin states with a cavity are only now succeeding. Here we report a two-spin-cavity system, achieved by embedding an InAs quantum dot molecule within a photonic crystal cavity. With this system we obtain a spin singlet-triplet Λ-system where the ground-state spin splitting exceeds the cavity linewidth by an order of magnitude. This allows us to observe cavity-stimulated Raman emission that is highly spin-selective. Moreover, we demonstrate the first cases of cavity-enhanced optical nonlinearities in a solid-state Λ-system. This provides an all-optical, local method to control the spin exchange splitting. Incorporation of a highly engineerable quantum dot molecule into the photonic crystal architecture advances prospects for a quantum network.