If I understood correctly from the YouTube letsplay, you can still give the artifact to the main guy on the git patrol by choosing the necessary dialogue options, but this causes an ultra hardcore battle with them.
You mean this one?
Im affraid he didnt "get it" to them ...
He simply presented the item, and then Kithrak fly off ... while the artefact remains in your inventory.
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.
I believe it will still reappear in your inventory, once story requires it ...
DIdnt try tho, but since that is how it work when you kill Shadowheart and dont loot it ... i presume it would be simmilar.