I presume that whoever owns the infrastructure will be able to report the losses and so reduce the GDP calculations. Damage to homes will probably also play a part in the calculations as their values are tracked by banks and the government. For smaller personal assets, I don't think the value would be tracked. Also, the 200B damages counting towards GDP could have been spent elsewhere, so the storm wouldn't really increase GDP. It would also reduce personal incomes of city dwellers.

As much as we like to shit on bourgeois economists, I think even they at least know better than to pull a broken window fallacy with GDP calculations, since that is something you are taught even in econ 101.

The real thing that (nominal) GDP measures is purchasing power of countries relative to other countries on the international markets at current prices (assuming frictionless trade). I do think it is a useful number to denote a few things, like, 1. is the country in a recession, 2. where does this country stand wrt imperialism, 3. how developed the country is.