Given a positive integer $n$, we wish to find the supremum of of the ratio of perimeter to diameter of convex n-gons, where the "diameter" is defined to be the length of the longest diagonal as usual.
First of all, it is known that if we allow $n$ to vary too, then the supremum is $π$. My conjecture for this problem is that the supremum is a maximum if we allow degenerate convex $n$-gons(those with less number of vertices than $n$), and that this "maximum" is reached by: a regular $n$-gon when n is odd; a regular $(n-1)$-gon when $n$ is even. (The even part of the conjecture is already proved to be false by RavenclawPrefect.)
The conjecture is false: for $n=4$, consider the polygon given by the vertices of a unit equilateral triangle, along with a point $1$ unit along the angle bisector of a vertex of the triangle.
This polygon has a ratio of
$$2+2\sqrt{2-\sqrt{3}}\approx3.03527$$
which improves on the ratio of $3$ attained by an equilateral triangle. (In general, constructions like this will be able to improve on an $(n-1)$-gon for even $n$.)
I'm not sure how to show this is optimal for $n=4$ (though I suspect it is), or to show that the regular polygons are optimal for odd $n$ (which I agree with you is likely the case).