  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 TaylorF2Ecc By using the Fisher
formalism and Bayesian inference, we 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. For a standard
thin viscous accretion disc around $M_z=10^5~{\rm M}_\odot$, $q=8$,
$\chi_{1,2}=0.9$, and $t_c=4$ years MBHB, we can confidently measure (with a
relative error of $<50 $ per cent) an Eddington ratio ${\rm f}_{\rm
Edd}\sim0.1$ for a circular binary and ${\rm f}_{\rm Edd}\sim1$ for an
eccentric system assuming ${O}(10)$ stronger gas torque near-merger than at the
currently explored much-wider binary separations. The minimum measurable
eccentricity is $e_0\gtrsim10^{-2.75}$ in vacuum and $e_0\gtrsim10^{-2}$ in
gas. 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.
