= Solution
For
$$
\mathbf U=z\widehat{\mathbf x},
\qquad
\Theta=z-\frac{\operatorname{Ro}}B y,
$$
both material derivatives vanish because the fields are independent of $x$. Since $\widehat{\mathbf z}\times\mathbf U=z\widehat{\mathbf y}$, steady momentum balance requires
$$
P_y=-z,
\qquad
P_z=\frac B{\operatorname{Ro}}z-y.
$$
These derivatives are compatible and integrate to
$$
\boxed{P=-yz+\frac{B}{2\operatorname{Ro}}z^2+\text{constant}}.
$$
The cross-stream temperature gradient and vertical stratification are in <thermal-wind balance> with the vertical shear.
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