= Solution
For a separated <harmonic function> $V=F(x)G(y)$, <Laplace's equation> gives $F''/F=-G''/G=-k^2$. Bounded periodic dependence in $x$ has $F=A\cos(kx)+B\sin(kx)$; decay on each half-plane selects $G=e^{-k|y|}$ for $k>0$, when the potential is matched continuously at the sheet. These are the separated decaying modes. General decaying solutions can be superpositions of them; decay alone does not force a single value of $k$.
The prescribed cosine density selects $k=1$ and $V=C\cos x\,e^{-|y|}$. Its normal <derivative> jump is $-2C\cos x$. Matching the stated jump $2\pi\Sigma_0\cos x$ gives
$$
\boxed{V(x,y)=-\pi\Sigma_0\cos x\,e^{-|y|}.}
$$
This is the <gravitational potential of a sinusoidal sheet> in the normalization of the question. It is continuous at the sheet and decays on both sides. Since $\Sigma_0\cos x$ changes sign, it can physically describe a Fourier component of a density perturbation rather than an everywhere positive isolated mass sheet.
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