Razed Ruins - Always, no matter what.
Gnarled Horn - Always, but I have Kannagi and solo/duo a lot, so the +50 AGI helps a lot. It's likely the best second atma choice for Blade: Jin users as well (I use it on Verethragna MNK and Victory Smite is a STR mod WS.)

The third atma depends on what I'm doing. If I'm in a DD situation fighting NMs and the like, which I often am, then Apoc almost always. Sometimes I replace this with Cloak & Dagger. I suppose if I was afking a lot then VV might be useful to have 100% TP at the start of every mob, but I can't foresee myself doing this very often. I do use Mounted Champion often for the Regen as it's nice when I'm solo /WAR (for red procs.)

Also, offhanding Kraken Club is questionable, at best. Here's a little napkin math.

The best case scenario for Kraken Club yields 1 hit per round mainhand 3.8 hits per round offhand. This would only ever occur if you were DNC subjob and using poor atmas, but we'll disregard that for the sake of this comparison. Using Atma of the Apocalypse, Twilight Belt, Brutal Earring, Epona's Ring, Double Attack from WAR subjob, and Atheling mantle yields 18% Triple Attack and 23% Double Attack. Assuming the distribution listed on studio gobli is correct, you're left with the following numbers of hits per round.

Code:
N	Kannagi	Club	Occurrence	Kannagi Hits	Club Hits
1	1	1	0.05		0.05		0.05
2	1	2	0.15		0.15		0.30
3	1	3	0.25		0.25		0.75
4	1	4	0.25		0.25		1.00
5	1	5	0.15		0.15		0.75
6	1	6	0.10		0.10		0.60
7	1	7	0.03		0.03		0.21
8	1	7	0.02		0.02		0.14
							
Total			1.00		1.00		3.80

Kannagi		Kamome
1.5486		1.5486
Finding the number of hits for Kannagi/Kamome was done by calculating Triple Attacks first and then calculating Double Attacks when Triple Attacks didn't proc (.18 * 3 + .82 * [ .23 * 2 + .77]). This is an assumption on how the game works, nobody really knows. Assuming Double Attacks overwrite Triple Attacks yielded a result within 2.75% of the one I listed.

Kannagi/Kraken Club total delay is 474 (1 round every 7.9 seconds, or 0.1266 rounds per second,) Kannagi/Kamome total delay is 400 (1 round every 6.66 seconds, or 0.15 rounds per second.) Ignoring haste/dual wield (will produce the same increase for both,) yields the following hits per second.

Code:
Kannagi		Kraken Club	Total
0.1266		0.4810		0.6076

Kannagi		Kamome		Total
0.2323		0.2323		0.4646
Factoring in delay and dual wield (assuming Iga Zukin +2, Ninja Chainmail +1, Suppanomimi and Rajas Ring) produces 4.6 TP per hit for Kannagi/Kraken Club and 4.5 TP per hit for Kannagi/Kamome. So you're left with 2.7950 TP/second on Kannagi/Kraken Club and 2.0907 TP/second on Kannagi/Kamome. Or, in otherwards, using the Kraken Club under optimal circumstances will only yield a 33.7% increase in TP gain.

Now, it's worth bringing up the very low DMG rating on Kraken Club. Assuming capped fSTRs for all weapons (which will definitely occur inside of Abyssea) leaves you with the following total DMG ratings.

Code:
Weapon		DMG	fSTR	Total DMG
Kannagi		56	14	70
Kamome		39	12	51
Kraken Club	11	9	20
You can factor this in with the hits per second to come up with a DMG/second number, which is odd but perhaps useful in this case for comparing melee DoT.

Code:
Kannagi		Kraken Club	Total
8.8620		9.6200		18.4820

Kannagi		Kamome
16.2610		11.8473		28.1083
So, going from Kannagi/KC > Kannagi/Kamome yields a 52.1% increase in melee DMG/second applied to the mob. This is a best case scenario for Kraken Club because it disregards Kannagi aftermath. Assuming you have the level 1 aftermath up 100% of the time, it's a 30% increase in DMG (on just the Kannagi.)

Code:
Kannagi		Kraken Club	Total
11.5206		9.6200		21.1406

Kannagi		Kamome		Total
21.1393		11.8473		32.9866
This yields a 56.0% increase in the same switch, and is the best case scenario for Kamome. So the observable number will fall between those two points.

I could provide some more napkin math on other facets of total damage output, but they're not going to mean much because those variables (pDIF, WSC, etc.) change too much from circumstances to provide useful ballpark figures. What you can conclude within a reasonable degree of confidence is this.

The increase in TP gain provided by offhanding Kraken Club is, at most, 33%. This won't always equate to 30% more weaponskills, however, because of things like TP overflow and holding TP because the mob has low health.

The increase in DMG/second by offhanding Kamome is around 51~56%. It's a reasonable measurement of optimal DoT from melee swings.

In this comparison, I ignored the effects that Kamome gives on critical damage. It's pretty big, that's the entire reason for offhanding Kamome. With RR & Qirmiz Tathlum you're looking at critical damage * 1.35 already. Kamome providing +0.10 to this yields a marginal gain of 7.41%. This increase would occur 74% of the time (with Gnarled Horn + Razed Ruins, 95% if you have Innin.) The actual DoT increase will be less than this, hard to say by how much without knowing a pDIF average.

I also made no analysis that the loss of 120 combat skill would have. You're losing a tremendous amount of accuracy and attack (108) on all of your offhand swings, which is the main source of your TP gain and, as it seems, melee DoT.

Lastly, your weaponskill damage will also be impacted by all of this. Hard to give good figures, but it would be significant. All in all, you are likely losing total DoT by using KC, not to mention unnecessary giving the mob more TP.