There is One way to get it completely out of your, or any other character's inventory - and that is to throw the character carrying it down a bottomless pit, so they warp to camp. Doing so causes all of their 'story' items to litter the edge of the hole you threw them down - including the lament configuration.
However, no matter what else you do, the puzzle box teleports to the character triggering the scene and binds to them immediately.
