We present microscopic many-body calculations for nonlinear two-dimensional coherent spectroscopy (2DCS) of trion and exciton polaritons in charge-tunable transition-metal dichalcogenide monolayers placed in optical microcavities. Charge tunability leads to a non-zero density electron gas that gives trions their brightness. It is a polaron quasiparticle formed by an exciton with a nonzero residue bound to an electronic gas. As a result, trione-polaritons are produced under intense light-matter coupling, as observed in recent experiments by Sidler \textit{et al.}.[}Nat. Phys. \textbf{13}, 255 (2017){]}. We analyze the structure of the trion polariton in detail by solving the extended Chevy hypothesis for the trion quasiparticle wavefunction. We confirm that the effective light-matter coupling of the trion-polariton is determined by the residues of the trion quasiparticle. The full many-body polaron state solution in Chevy ansatz allows microscopic computation of the nonlinear 2DCS spectra of both trion and exciton polaritons. We predict the presence of three types of off-diagonal cross peaks in the 2DCS spectra as an indication of the coherence between different branches of the trion and exciton polaritons. Due to the sensitivity of 2DCS spectra to quasiparticle interactions, our work provides a good starting point for investigating the strong nonlinearity exhibited by trion-polaritons in some recent exciton-polariton experiments.
