This paper is dedicated to extending the focusing$\slash$defocusing Calogero--Moser--Sutherland cubic derivative Schr\"odinger equation (CMSdNLS), introduced in G\'erard--Lenzmann [arXiv:2208.04105] and Badreddine [arXiv:2303.01087, arXiv:2307.01592], to a system of two matrix-valued variables, leading to the following intertwined system, \begin{equation*} \begin{cases} i\partial_t U + \partial_x^2 U = - \tfrac{1}{2} U \left(\mathrm{D} + |\mathrm{D}| \right) \left( V^* U\right) - \tfrac{1}{2} V \left(\mathrm{D} + |\mathrm{D}| \right) \left( U^* U\right),\\ i\partial_t V + \partial_x^2 V = - \tfrac{1}{2} V \left(\mathrm{D} + |\mathrm{D}| \right) \left( U^* V\right) - \tfrac{1}{2} U \left(\mathrm{D} + |\mathrm{D}| \right) \left( V^* V\right),\\ \end {cases} \quad \mathrm{D}= -i\partial_x. \end{equation*}This system still enjoys a Lax pair structure, which allows to establish an explicit formula for general solutions on the torus and on the line. As a consequence, this system can be considered as an integrable perturbation of both the linear Schr\"odinger equation and the CMSdNLS equations.