I occasionally lose track of items on my retainers. If you're positive you threw it away, and it isn't just lost somewhere in your retainers' inventories, then yeah, you're gonna have to farm another. If you think it might just be lost, you can run a /isearch for it. If you're unfamiliar with the isearch command, you can find items in this way:

1. Type "/isearch Deathbringer". This will bring up a list of all items in the game that have "Deathbringer" within the name - including Animated Deathbringer.
2. Right click on "Animated Deathbringer" in that list of items, and choose "Search for items" from the menu that appears. This will tell you how many Animated Deathbringers you have, and what tabs of which inventories those Animated Deathbringers can be found in. (Since it's an exclusive item, obviously there will only be one or zero Animated Deathbringers in your possession.)

This will verify for certain whether your Animated Deathbringer was thrown away or not.

If you discover that yes, your Animated Deathbringer is gone, you do have one other alternative to grinding crystals again. The GMs will restore one lost item for you, if you accidentally throw it away. The kicker is that they will only perform this service for you once, ever. Personally, grinding crystals isn't THAT bad, and I'd save your one-time item restore for something more difficult to replace, but the option is there!

From the FFXIV Support Center

Items sold to an NPC can be bought back. The last 10 items sold to an NPC are displayed on that NPC's buy-back list.
However, if you have switched zones or logged out after selling the item, the NPC's buy-back list will be deleted and the item cannot be bought back.

Items sold to an NPC can be recovered using the above methods, but items lost for any other reason cannot be recovered.

However, there is a possibility that an item that has been "dropped," "consumed," or "converted to Materia," can be restored through the item recovery service. This service is only offered once per Square Enix account.

If you would like to request a recovery, please contact us within 1 week of losing the item and provide specific information, such as item name, time lost, area where the item was lost, etc.

Following a GM investigation, if it is confirmed that an item can be restored, the item will be restored. The results should be sent after several days, but the investigation may take several weeks.

However, in any of the below situations, we may not be able to restore the item. If there is any confusion about whether a request falls into these categories, or if there is anything that is unclear, please discuss the matter with a GM.

・The request was made 1 week or more after the item was lost.
・A GM investigated but could not find any log evidence.
・You have requested an item recovery before.
・The item had materia attached or contained unique information
　*It may be possible to restore the item in its initial condition