Alter the container icon to reflect its 'fullness'?
 

Would it be possible to alter the bag, pack or strongbox icon to represent how full it is?

I wondered if there's something like the spell / ability recast counter, where it gradually fills from a darkened icon.

Transitional might not be possible, but maybe a transparent -X- overlay on full containers could be useful.

I suppose this depends whether or not this info is being pushed down from the server... Or isn't 'queried' until the user clicks on the container.

The info isn't available.