Had trouble with this as I didn't have all pets. Based on comments above, I used random pet first, Fel Flame (1,1,1) and Dark Phoenix Hatching (2.2.2)
First pet dies
Phoenix in - Laser x 4, Dark rebirth and then laser again
Fel flame comes in - immolate, conflagrate and burn x 3
That worked for me.
I didn't have all the pets here, so I used a random elemental first, then Fetid Waveling (1,2,2) and Molten Corgi (2,1,1). Just use Corrosion, then Ooze Touch until one round before Fetid dies, then use Corrosion again. Use Puppies of the Flame before the big attack then Flamethrower.
I did this with a P/P Crimson Geode, as it's far more powerful than Ele Geode, and a Fetid Waveling in place of the Tainted, using Corrosion instead of Acidic Goo which works just as well if you use; Corrosion > Ooze Touch > Ooze Touch > Corrosion. It's a tiny bit down on damage compared to Acidic Goo > Ooze Touch x 3, but it's a negligible amount.
The mid-battle comment in this strat doesn't make sense: "If Tainted Waveling dies in Turn 6 and Earseeker has 2 rounds left on Antaen Cannon swap to Elementium Geode and use Amplify Magic before Turn 7" - to do this then carry out the action in Turn 7, you need to swap to the Elementium Geode (one round) then the Tainted Waveling (one round) - both in one round...
Works perfectly if your Tainted Waveling doesn't die, but you need to re-work what you're suggesting for rounds ~6-7+
It doesn't take a turn when swapping in a live pet for a dead one. If the Waveling dies on round 6, your Geode will be in situ on round 7 with 2 turns to go until Antean Cannon comes off cd. In that situation, you should use Amp Magic then Feedback before swapping to SotF to soak the Cannon. This is because, as always, there's a chance of a crit, and a crit from one buffed Feedback is better than one crit from 2 unbuffed Feedbacks.
Soul of the Forge (1,1,1)
Fel Flame (1,2,1)
Corehound Pup (X,2,1)
Round 1 - Start with SotF, use flamethrower. If he dies in first round restart. It's important to have this guy alive because at the end of the fight, sometimes core hound pup will die to poison depending on RNG.
Round 2+ - swap to Fel flame, and maximize dps (I think I used Scorched Earth, Conflag, Burn, Burn, Burn).
Once Fel Flame dies bring in Core Hound Pup, and use Dodge when Antean Cannon has 1 round CD. Then just burn burn burn!