Schrödinger cat states provide a hardware-efficient platform for bosonic quantum error correction by encoding logical information in protected manifolds of harmonic oscillators. While previous work has demonstrated the dissipative stabilization of multi-mode Schrödinger cat states as robust quantum memories, a framework for universal quantum computation has remained unavailable. Here we work towards this by introducing a universal gate set for dissipatively stabilized multi-mode cat qubits as a necessary resource for computation. Using a chain of Kerr non-linear oscillators coupled through engineered non-local dissipation and an effective low-dimensional description, we show how single-qubit control can be achieved through arbitrary rotation around the $X$-axis and $\pi /2$-rotation around the $Z$-axis. We further show how coupling two such stabilized arrays through just one oscillator on each respective array enables coherent entangling operations through implementation of the $X{X}_{\pi /2}$ gate. Numerical simulations demonstrate high-fidelity gate dynamics and entanglement generation under realistic parameters. Finally, we analyze the effects of induced and intrinsic photon loss, disorder, and the validity regime of the effective low-dimensional theory. Our results establish dissipatively stabilized multi-mode Schrödinger cat states as a potential architecture for universal bosonic quantum computation.