= Solution
The <potential-vorticity gradients in a meridional two-layer current> are important here: besides $\bar q_{n,y}=\beta$, one has $\bar q_{1,x}=-FV$ and $\bar q_{2,x}=FV$. They must be retained when linearizing, even though the basic <relative vorticity> vanishes.
Let $K^2=k^2+l^2$ and $A=K^2+F$. For a <normal mode> the <two-layer quasi-geostrophic potential vorticity> amplitudes are
$$
\widehat q_1=-A\widehat\psi_1+F\widehat\psi_2,\qquad \widehat q_2=F\widehat\psi_1-A\widehat\psi_2.
$$
Because the basic velocities are northward $(V,0)$ in the two layers, their advection frequencies are $\sigma-lV$ and $\sigma$. The uniform forcing has no perturbation, $W'=0$. The linear equations are therefore
$$
(\partial_t+V\partial_y)q'_1+FV\psi'_{1,y}+\beta\psi'_{1,x}=0,
\qquad \partial_tq'_2-FV\psi'_{2,y}+\beta\psi'_{2,x}=0.
$$
In particular, a perturbation's zonal velocity advects the basic interface-induced zonal <potential-vorticity gradient>. Substitution of the <plane wave> gives
$$
\begin{pmatrix}
A(\sigma-lV)+\beta k+FlV&-F(\sigma-lV)\\
-F\sigma&A\sigma+\beta k-FlV
\end{pmatrix}
\begin{pmatrix}\widehat\psi_1\\\widehat\psi_2\end{pmatrix}=0.
$$
A nonzero disturbance exists exactly when the determinant vanishes. The requested relation for <linear stability of a meridional two-layer current> is
$$
\boxed{[A(\sigma-lV)+\beta k+FlV][A\sigma+\beta k-FlV]-F^2\sigma(\sigma-lV)=0.}
$$
Equivalently, with $\mathcal B=K^2(K^2+2F)$,
$$
\boxed{\mathcal B\sigma^2+[2A\beta k-\mathcal B lV]\sigma+\beta^2k^2-A\beta klV+FK^2l^2V^2=0.}
$$
Its discriminant is
$$
\Delta_\sigma=4F^2\beta^2k^2+\mathcal B K^2(K^2-2F)l^2V^2.
$$
For $K>0$, exponential <baroclinic instability> occurs when this discriminant is negative; otherwise the two frequencies are real. This also displays the stabilizing contribution of the <planetary vorticity> gradient when $k\ne0$. As a check, $V=0$ gives the uncoupled <barotropic mode> and <baroclinic mode> frequencies $-\beta k/K^2$ and $-\beta k/(K^2+2F)$.
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