Is there a closed form expression for main-lobe width increase given a window? We know that when we window a signal, we increase the main-lobe width. Let 'main-lobe-width' here be the null-to-null bandwidth of the main lobe. Let us further more say that the main-lobe width of a square window is '1'.

What I would like to know, is given a particular symmetric window, is there a closed form solution for how much the main-lobe will increase by, relative to that of a square window?

I can look up this percentage increase for various windows just fine by the way. I am asking if there is a closed form solution for what the percentage increase is, given an arbitrary symmetric window.
If I understand you correctly, you are in fact asking for a closed-form solution for the position of the first zero of a given window function. Such a solution cannot exist, because what you would need is an analytic solution to a polynomial equation, which does not exist for polynomials of degree 5 or higher (that's the Abel-Ruffini Theorem).