You will need a toy called Moroes' Famous Polish.
This toy can be obtained from the quest Feeling A Bit Morose.
To get the quest you will need a Lunarfall Inn / Frostwall Tavern in your WoD Garrison.
Only a rank 1 Inn / Tavern should be required to get the quest. It is random which quest you have in the Inn / Tavern everyday, so you just have to wait until Moroes shows up and do his quest.
Once you have the Moroes' Famous Polish toy and inside the Battle of Dazar'alor raid, in a part of the raid where your character has changed to its opposite factions counterpart:
1. Use the Moroes' Famous Polish toy
2. Remove the buff you gain from the toy by right clicking the buff.
3. Enjoy your own transmog while raiding as the opposite faction. Lasts until death and the toy has a 30 min CD, so use it when you know you're not going to die for a while =)
Hope this was helpful for atleast some of you.