= Solution
Use $(\bar u,\bar v)=(-\bar\psi_y,\bar\psi_x)$ and the hydrostatic <thermal-wind balance> convention $\bar b=f_0\bar\psi_z$. Because the basic state is independent of $x$, $\bar v=0$. The thermal-wind relation is
$$
f_0\bar u_z=-\bar b_y=-A'(y).
$$
Integration from the resting lower boundary gives
$$
\boxed{\bar{\mathbf u}(y,z)=-\frac{zA'(y)}{f_0}\widehat{\mathbf x}.}
$$
A compatible <quasi-geostrophic streamfunction> is
$$
\bar\psi(y,z)=\frac1{f_0}\left[zA(y)+\int_0^zB(s)\,ds\right]+C,
$$
where $C$ is a spatial constant. Its horizontal <Laplacian> is $zA''(y)/f_0$ and its second vertical derivative is $B'(z)/f_0$. Substituting into the given <three-dimensional quasi-geostrophic potential vorticity> yields
$$
\boxed{\bar q=f_0+\frac{zA''(y)}{f_0}+\frac{f_0}{N^2}B'(z).}
$$
In particular, the horizontal-curvature term has a plus sign: $\bar\zeta=-\bar u_y=zA''/f_0$.
The <reference-buoyancy convention in quasi-geostrophic potential vorticity> should be stated. If $\bar b$ includes the full reference profile $N^2z$, the formula above includes its constant contribution $f_0$. If instead the <quasi-geostrophic streamfunction> represents pressure relative to that reference hydrostatic state, use $\bar b-N^2z=f_0\bar\psi_z$. Then the quoted $\bar q$ is reduced by the constant $f_0$. Both conventions give exactly the same velocity and <potential-vorticity gradients>, and therefore the same subsequent dynamics. The coefficient $N$ in the stated linear model is a prescribed reference <buoyancy frequency>; if it is also required to equal the full physical buoyancy frequency everywhere, then $B'(z)=N^2$, rather than an arbitrary function. The formal expression above retains the requested arbitrary $B$ with the stated constant coefficient.
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