Rowlett is hosting The Sound of Symmetry here. The proof of Theorem 4 is exactly as Noam Elkies suggests: Via the Dirichlet heat trace's asymptotic expansion, both area and perimeter are determined by the spectrum, and so for any $n$-gon $\Omega$ the isoperimetric ratio $|\Omega|/|\partial\Omega|^2$ is determined by the spectrum. The content of the proof is that this ratio is globally maximized among $n$-gons at the regular $n$-gon. This determines the regular $n$-gon and implies that any $n$-gon isospectral to the regular $n$-gon is in fact isometric to it. Unless there's a point in the proof where you're confused, I think this settles the question.

(I have a suspicion that the third coefficient of the heat trace in convex polygons $$ \frac{1}{24}\sum_{\mbox{angles }\alpha_i} \bigg(\frac{\pi}{\alpha_i} - \frac{\alpha_i}{\pi}\bigg) $$ is also extremal at the regular $n$-gon. This may be another way to show the desired result.)

Rowlett is hosting The Sound of Symmetry here. The proof of Theorem 4 is exactly as Noam Elkies suggests: Via the Dirichlet heat trace's asymptotic expansion, both area and perimeter are determined by the spectrum, and so for any $n$-gon $\Omega$ the isoperimetric ratio $|\Omega|/|\partial\Omega|^2$ is determined by the spectrum. The content of the proof is that this ratio is globally maximized among $n$-gons at the regular $n$-gon. This determines the regular $n$-gon and implies that any $n$-gon isospectral to the regular $n$-gon is in fact isometric to it. Unless there's a point in the proof where you're confused, I think this settles the question.

(I have a suspicion that the third coefficient of the heat trace in convex polygons $$ \frac{1}{24}\sum_{\mbox{angles }\alpha_i} \bigg(\frac{\pi}{\alpha_i} - \frac{\alpha_i}{\pi}\bigg) $$ is also extremal at the regular $n$-gon. This may be another way to show a version of the desired result for convex $n$-gons.)