Bernoulli is a reasonable approach so long as the gas flow is reasonably 'gentle' (no vortices or compression). However, to compare two cross sections of the exhaust then you need to integrate across the cross sections, yielding $$ \left(\frac{v_1^2} {2} + \frac{p_1} {\rho} \right) A_1 + \int_{A_1} gz dA = \left(\frac{v_2^2} {2} + \frac{p_2} {\rho} \right) A_2 + \int_{A_2} gz dA $$ The gravity term depends on the orientation and shape of the exhaust, but can probably be ignored to this level of approximation for a gas.

Bernoulli is a reasonable approach so long as the gas flow is reasonably 'gentle' (no vortices or compression). However, to compare two cross sections of the exhaust then you need to average across the cross sections, yielding $$ \frac{v_1^2} {2} + \frac{p_1} {\rho} + \frac{1} {A_1} \int_{A_1} gz dA = \frac{v_2^2} {2} + \frac{p_2} {\rho} + \frac{1} {A_2} \int_{A_2} gz dA $$ The gravity term depends on the orientation and shape of the exhaust, but can probably be ignored to this level of approximation for a gas.