the sun - The relation between the light of full moon and the distance between the sun and the earth

The strength of the light from the Sun scales with the inverse square of distance [note 1]. That means that we would need to have the Earth-Moon system at $frac{1}{sqrt{2}}$ (approx 0.7) AU for the full Moon to be twice as bright [note 2].

Note 1: The fact that the Sun is not a point source of light has only a very minor effect on the scaling.
Note 2: Technically, it will be slightly brighter because more of the absorbed radiation will be radiated away in the visible spectrum. This is however also a negligible effect.