[beethove]

If the radical is on chlorine, Formal charge is 0 = -1, -1; CL = 2

But if the radical is on one atom of oxygen and there's a double bond on the other oxygen, then FC is O (double bond) = 0, O(radical) = 0; Chlorine = 0. The FC difference is less than the former structure. Shouldn't this be the most stable structure?

[Corribus]

Formal charge is really only a bookkeeping device. The general rule of thumb is to rely on electronegativity differentials to decide where to place negative charges. In reality, things are not so simple and charges are delocalized to some extent (e.g., in molecular orbitals, if you're willing to use this model).