The Moon is defined as full when its ecliptic longitude is 180° distant
from the Sun's.

But what is the cycle for when the maximum proportion of the Moon's
apparent area is sunlit, and for observers at what latitudes of the Earth's
surface is this maximum realised?

Note that the plane of the Moon's orbit around the Earth is inclined at an
angle of 5° to the plane of the Earth's around the Sun. To get a handle on
why this affects the Moon's apparent fullness, imagine if it were 90°.
Given the inclination I am sceptical as to whether the proportion ever
reaches 50%.