This was a hard one, but now with the items needed for the quest available during most of the Log-in Campaigns it makes it much easier. You can get the items quickly using your login points or if you are not participating in the Log-in Campaigns you still can farm the items you will need for this limit break quest. I've attached a walk thru of the quest below as well as source pages for the items you will need. If you have to farm the items the mob info is provided on the source page links.
The main item you will need to get is a "Seasoning Stone" to obtain the Key item "Soul gem clasp" which you receive by trading the Seasoning Stone to the Nomad Moogle next to Maat in Ru'Lude Gardens.
BUT FIRST, to make the Limit Break battle easier, you will need get another Seasoning Stone, Fossilized Fang and Fossilized Bone which you will trade to Degenhard in Bastok Markets at I-10 to receive an "Olde Rarab Tail". You use this item during the fight with Atori-Mouri and he will freeze and lose his damage resistance. You will have to time the use of this item correctly to get the best effect.
See the Links below for additional help, I've included a link to a testimonials/strategy page link for the fight itself that has been written by other players that you may find very helpful. When I've done the fight I've had the best luck using 3-4 Beastmasters with Gooey Gerard, a jug pet summoned with Cloudy Wheat Broth. But you might find a set up that works best for you...and don't be shy about shouting for help. Eventually you will find some level 99s that will help you. Good luck. Skyfiredancer (talk) 08:26, August 11, 2015 (UTC)
Limit Break - http://ffxiclopedia.wikia.com/wiki/Beyond_Infinity
Seasoning Stone - http://ffxiclopedia.wikia.com/wiki/Seasoning_Stone
Testimonials page - http://ffxiclopedia.wikia.com/wiki/BCNM_Atori-Tutori_%3F%3F%3F_Stategies
Fossilized Fang - http://ffxiclopedia.wikia.com/wiki/Fossilized_Fang
Fossilized Bone - http://ffxiclopedia.wikia.com/wiki/Fossilized_Bone
Olde Rarab Tail -http://ffxiclopedia.wikia.com/wiki/Olde_Rarab_Tail