Badge of Vows 83 is a quest item from the From A Different Sky - Part 9 DLC. Merely picking it up satisfies the goal needed to complete the quest.
As any of the magick using vocations, you need to cast Frigor / High Frigor near the shelf. You need to create an ice platform to stand on so you can pick up the badge. Considering you will likely be assaulted by soldiers while doing this, it will be difficult, but it can be done.
You can temporarily thin out their numbers by running into the dining room upstairs and eliminating all the soldiers. Additionally, it is recommended that you take a fully party of pawns that are capable of keeping the soldiers busy.
Alternate method: This badge can be obtained by spamming the Sunflare ability (Magick Archer). Stand directly in front of the book shelf and point your camera straight at it. Bring something to recover stamina because this will take many attempts. Keep using your Sunflare ability until the badge falls from the shelf.
Explosives and explosive spells, such as Throwblast or High Ingle, can be used to dislodge this badge from its perch, as well, but it will take a lot of time to do so.
Warning: This quest should be done before, or during, the quest "A Warm Welcome." After the quest is completed, you will have a much tougher time acquiring this badge due to guard interference.
Note: The same method could, theoretically, be used to acquire Badge of Vows 77.