We assess the possibility of detecting both eccentricity and gas effects (migration and accretion) in the gravitational wave (GW) signal from LISA massive black hole binaries (MBHBs) at redshift $z=1$. Gas induces a phase correction to the GW signal with an effective amplitude ($C_{\rm g}$) and a semi-major axis dependence (assumed to follow a power-law with slope $n_{\rm g}$). We use a complete model of the LISA response, and employ a gas-corrected post-Newtonian in-spiral-only waveform model \textsc{TaylorF2Ecc}. By using the Fisher formalism and Bayesian inference, we explore LISA's ability to constrain $C_{\rm g}$ together with the initial eccentricity $e_0$, the total redshifted mass $M_z$, the primary-to-secondary mass ratio $q$, the dimensionless spins $\chi_{1,2}$ of both component BHs, and the time of coalescence $t_c$. We find that simultaneously constraining $C_{\rm g}$ and $e_0$ leads to worse constraints on both parameters with respect to when considered individually. Assuming a standard thin viscous accretion disc, for $M_z=10^6~{\rm M}_\odot$, $q=8$, $\chi_{1,2}=0.9$, and $t_c=4$ years, we can confidently measure (with a relative error of $<50 $ per cent) an Eddington ratio as small as ${\rm f}_{\rm Edd}\sim0.1$ for a circular binary while for an eccentric system only ${\rm f}_{\rm Edd}\gtrsim1$ can be inferred. The minimum measurable eccentricity is $e_0\gtrsim10^{-2.75}$ in vacuum and $e_0\gtrsim10^{-2}$ in the presence of a circumbinary disc. A weak environmental perturbation (${\rm f}_{\rm Edd}\lesssim1$) to a circular binary can be mimicked by an orbital eccentricity during in-spiral, implying that an electromagnetic counterpart would be required to confirm the presence of an accretion disc.
Comment: 14 pages, 8 figures. Submitted to MNRAS