A Big Problem with the New Summoner.

[Dyvid]

So after looking at it the basic fight will look like this:
Miasma and Bio -> Fester -> Ruin Ruin Ruin -> Fester -> Ruin Ruin Ruin -> Fester -> DWT -> Tri-D -> Ruin Spam -> Death Flare -> AF -> {Repeat Rotation} -> Miasma and Bio -> Bahamut. Add Shadow Flare somewhere into the mix.

Now Pets may or may not be tricky depending on how you play it. You could just summon Garuda and keep her on Sic the whole time. Or you could try swapping Ifrit and Garuda back and forth for maximum debuff (Radiant Shields and Contagion*) but this will create a challenge because of how Egi skills are designed. All Egi's share the same ability slots, 3 4. Radiant Shield is on 3 and Contagion* is on 4. So if on Sic, Garuda's will automatically use Aerial Slash (Slot 3) and put it on a cooldown. If you summon Ifrit at this time Radiant Shield will be on a 30s cooldown because Garuda used Slot 3's ability. Ideally you would summon each every 30s to keep the debuffs up.

In other words you'll have to keep both pets on Obey and manually use abilities while keeping the timers in mind. Whether this is actually workable or just a dps lose I can't say until actually used.

[Semmi]

It will likely work out to be a DPS loss, but I'm not an expert on pet mechanics and don't know all of the quirks they have in detail. Even if it were a gain, it wouldn't be sustainable given the MP costs.

[ShaolinMike]

So after looking at it the basic fight will look like this:
Miasma and Bio -> Fester -> Ruin Ruin Ruin -> Fester -> Ruin Ruin Ruin -> Fester -> DWT -> Tri-D -> Ruin Spam -> Death Flare -> AF -> {Repeat Rotation} -> Miasma and Bio -> Bahamut. Add Shadow Flare somewhere into the mix.

You can open with Ruin II (Tri-Disaster) since the cooldown gets reset when you enter DWT. Might also need to replace 1 or 2 Festers with Painflares to get into DWT quicker and synergize with other party member buffs in their opener. But yeah this appears to be the basic idea.