If the tech requirements are met and the weather is at Cat II minima, I'd look pretty foolish doing a go-around because of a temporary deterioration as I got to minima, when I could have done the Cat III. As an aside, my company require us to autoland from a Cat II approach unless there is a tech problem that prevents it. So, if the aircraft tells me it's Cat III capable, and the ILS/Airport meet the requirements, I'll be in the bar shortly after.......
Would you go below minima on a cat 1 approach? or non precision?
You might be at the bar first but at least the NAA guy will know where to find you when he sees you made the approach below cat II minima when cat II was all that was declared by the airfield.
It's like saying my ac is certified cat II autoland but I flew cat III minima. You simply can't do it legally.