Weight itself has a potential effect because Vmcg demonstration is a dynamic manoeuvre, not a static balance. So the actual inertia of the aircraft (both mass and moments of inertia) have some bearing on the behaviour of the aircraft.
Crudely, a heavier aircraft will respond less abruptly to the engine failure (which will tend to reduce the lateral displacement for a given case) but will also respond less to the pilot action to restore the aircraft to centreline (thus tending to make the displacement worse). Which is the more significant is a fine balance - to the extent that it may be not a practical concern.
There may also be design features that cause there to be a weight effect - thinking aloud, maybe tyre characteristics that cause the mainwheel moemnt generation to be different at heavier than light weights due to overloading the tyres, such that you get slipping/skidding at heavier weights?