http://forum.square-enix.com/ffxiv/threads/155891-FAQ-Zoadiac-Weapons-You

of particular note:

Q: Do I need to have my Zenith Relic Weapon equipped? Also do I have to be on the Job that I originally accepted the quest on?

A: YES, you must have your Zenith Relic Weapon equipped while doing these fates. The exact text comes from a System Message given after you talk to Jalzahn and it states:

System:: By participating in certain FATEs while equipped with a relic weapon zenith, you may obtain an atma crystal.

If you want the full dialogue go here and click on the "Dialogue" tab: http://xivdb.com/?quest/66971/Up-in-Arms

However, it has been confirmed that you DO NOT need to be on the same job that you originally accepted the quest on. For example if you started the quest on a White Mage, you can still obtain Atma by switching to your Bard who also has their Zenith Relic Weapon equipped and obtain the items.

and

Q: I have multiple classes with Zenith Weapons, will I have to be on the job that started the original quest, or can I choose which one to upgrade? i.e. If I started the quest on White Mage, but want to upgrade my Bard Zenith Weapon instead, can I do it?

A:
Shirai's Avatar
Quote Originally Posted by Shirai View Post
Once you recieve all 12 Atmas you can choose which relic you want to upgrade.
Just be sure not to be on that job or have a different weapon equiped when you do as you "hand in" your weapon.

I almost always farmed on WAR as opposed to BLM, particularly in lower level zones. Flash and Overpower are great agro grabs and rank you higher in fates. Not that the rating is so important,but being able to get gold in less time means maximizing your reward while freeing you up to bounce to another available fate.

This is all about the 2.5% drop rate. Go in w/ the job that allows you to claim/kill the fastest, and move to the next fate. More fates = more chances = more sanity. :)