The controlled generation of nonclassical light beyond single photons remains a central challenge in quantum optics, due to the difficulty of enhancing multiphoton processes while suppressing lower-order excitations. Here we propose an interference-interaction-engineered scheme for programmable few-photon emission in a cavity-QED system of three atoms coupled to orthogonal cavity modes. By adiabatically eliminating an auxiliary Fabry-P'erot cavity, we generate a tunable cavity-mediated spin-exchange interaction $\chi$. Combined with a controllable position-dependent phase $\varphi$, it reshapes the many-body dressed-state spectrum. Phase-controlled interference selectively addresses different excitation manifolds, whereas the cavity-mediated SEI enhances the spectral separation between neighboring manifolds, thereby stabilizing the selected excitation manifold and the corresponding photon-emission channel. For $\varphi =0$, constructive interference favors the low-excitation manifolds, yielding tunable single- and two-photon emission associated with the $N=1$ and $N=2$ manifolds. In contrast, for $\varphi =2\pi /3$, destructive interference suppresses lower-order excitation pathways and activates a resonant three-photon channel originating from the $N=3$ manifold. Importantly, the cavity-mediated interaction $\chi$ further reinforces the spectral isolation between neighboring manifolds, enabling a substantial improvement in multiphoton purity while maintaining a sizable photon population. We demonstrate a three-order-of-magnitude enhancement in two-photon purity and more than two orders of magnitude improvement in three-photon emission. Our results establish a unified interference-interaction framework in which effective optical nonlinearities can be programmably engineered through phase-controlled interference and cavity-mediated interactions, providing a scalable route toward high-purity multiphoton sources and programmable quantum photonic devices.