Other effects I was considering.
Is the bottle rotationally symmetric? Is there say a weight of congealed shampoo in it?
If there was a slight tilt in this whole setup, the bottle could be marginally off vertical. Empty, the centre of gravity is quite high above the bar, and the slight tilt puts it slightly inward. With some water in the centre of gravity is lower. Full to the brim, the centre of gravity isn’t much lower.