Many natural and industrial processes involve the flow of fluids made of solid particles suspended in Non-Newtonian liquid matrices, which are challenging to control due to the fluid's non-linear rheology. In the present work, a Taylor-Couette canonical system is used to investigate the flow of dilute to semi-dilute suspensions of neutrally buoyant spherical particles in highly elastic base polymer solutions. Friction measurement synchronized with direct flow visualisation are combined to characterize the critical conditions for the onset of elasto-inertial instabilities, expected here as a direct transition to Elasto-Inertial Turbulence (EIT). Adding low particle volume fraction (≤ 2%, dilute regime) does not affect the nature of the primary transition and reduces the critical Weissenberg number for the onset of EIT, despite a significant decrease in the apparent fluid elasticity. However, for particle volume fractions ≥ 6% (semi-dilute regime), EIT is no longer observed in the explored Reynolds range, suggesting an apparent relaminarization with yet not further decrease in fluid elasticity. Instead, a new regime termed here Elasto-Inertial Dissipative (EID), was uncovered. It originates from particle-particle interactions altering particle-polymer interactions and occurring under elasto-inertial conditions comparable to that of EIT. Increasing particle volume fraction in the semi-dilute regime and, in so, the particle contribution to the overall viscosity, delays the onset of EID similarly to what was observed previously for EIT in lower elasticity fluids. After this onset, a decrease in the pseudo-Nusselt number observed with increasing inertia and particle-to-polymer concentration ratio confirms a particle-induced alteration of energy transfers in the flow.