A1S: Most of the fight is a Striking Dummy, so this is where knowledge of DRG fundamentals is going to be essential. Normal offensive cooldowns (BFB, IR, potion) should be used as soon as possible, Geirskogul needs to be managed properly (3 per minute), Invigorate needs to be popped at ~450 TP during phase one, and Blood of the Dragon--as is always true--should be active at all times.
Beyond that, the key to maximizing DRG DPS in this fight lies in adds management. More specifically, for each set of adds, your dragoon should land at least one Geirskogul on both the boss and the add he is handling. Additionally, adds should be dealt with in a way that maximizes uptime. He can either burn the add down on its laser position before tether; or tag the add with one hit, return to the boss for 2-3 GCDs until Resin goes out, and then kill the add.
Finally, keep in mind that your DPS output for this fight will vary dramatically depending on how long the encounter lasts. Even the best of dragoons will achieve much lower numbers in a 9-minute clear than they will in a 4-minute clear.
A2S: Chaos Thrust combo and Geirskogul while maintaining Heavy Thrust. It's that simple. His CT combo will be superior to his FT combo as long as the add lives for 3 seconds or longer beyond the application of Chaos Thrust. Doom Spike might be worth it in a few situations if you're getting a Goad or two throughout the encounter. If he's going to use Doom Spike, it'll be most appropriate for waves 3 and 8.
If he doesn't get any Goads at all, then it's likely that he'll run out of TP by the end of the encounter, even without using any Doom Spikes. If so, he should request BRD/MCH TP regen as necessary, and/or accept some downtime during wave 4.
One important part of the fight for DRGs is wave 7, consisting of the 4 Jag'd Dolls. I would strongly encourage your tanks to group them relatively close together, if they aren't already. This allows your dragoon to Geirskogul all four adds numerous times under very friendly conditions in terms of uptime and cooldown management.
As for cooldown management, I'm typically popping BFB first in wave 3, then using it more or less as soon as it next becomes available. I use Battle Litany at waves 3, 6, and 8; however, BL timing may vary depending on how quickly your group can clear the encounter.