They key to this fight is following the following priority list:

Immolate > Conflagrate > Burn

Basically, make sure the target is immolated at all times (it lasts 4 rounds), use Conflagrate on cooldown, and Burn the rest of the time. Note that on Corporal Hammer, you should get 2 conflags on him due to his pet swapping attack that he spams. Outside of Corporal Hammer's attack, you should not have to swap pets or be swapping pets except through death. Good luck.

I used the above strategy and priorities with a Dark Phoenix Hatchling 1,2,1 and 2 Fel Flames 1,1,1. The Dark Phoenix took out Bert's first pet and a chunk of the second. The weakened Fel Flame finished off the second pet, and got a hit on Clamp before it died. The second Fel Flame finished off Clamp with approximately half health left. Thank you for the strategy!

This works 100% of the time

DPH = Dark Pheonix Hatchling
PFS= Pandaren Fire Spirit
PH= Pheonix Hatchling

R1: DPH: Immolate
R2: PFS: Conflagrate
R3: DPH: Conflagrate
R4: PFS: Burn
==The pandaren Fire spirit is up, Corporal Hammer died, M-37 Enters==
R5: Immolate
R6: Burn
R7: Conflagrate
==M-37 Dies , PFS Dies ==
==Enter with PH==
R8: Immolate
R9: Conflagrate
R10: Burn
R11: Burn, PH dies
R12: DPH: Burn

thanks this worked great for me

Didn't have either a Searing Scorchling or a Dark Phoenix Hatchling. Substitute Scorchling with 1/2/1 Ominous Flame, and Phoenix with 2/2/1 Crazy Carrot. I put the Fel Flame in the first spot, the Ominous Flame second, and Crazy Carrot third.

Started by immolating Hammer, then conflagrated with Ominous Flame as it got switched in, then conflagrated with Fel Flame, and then dropped Scorched Earth to ensure the enemy would always be ablaze (whole team is elementals, so it does nothing to me).

Once M-37 came in, I conflagrated, immolate/scorched earth, or burned as necessary until my pet died. As soon as it was dead, I sent in the Carrot. Ironbark, then blistering cold, and then kept using Ironbark until M-37 was fully dead.

Clamp comes in. Started by using Aged Yolk to remove the fire debuffs, then Ironbark, and then blistering cold, after which I continued to use Ironbark until Clamp was dead. Crazy carrot survived with some 200 HP left, though I did get a couple of crits which may have influenced my win chance.

All in all, I had no real strategy planned and succeeded with this set up on first try. Do give it a try if you don't have above mentioned pets but have these two.

This worked great! Thanks for the suggestion :)

May I suggest instead of the Dark Phoenix Hatchling swapping to the Crazy Carrot or another like it? Still elemental, but both Aged Yolk and Ironbark are really helpful for dealing with the fallout from M-37, which seemed to be the tipping point for this fight. Aged Yolk can remove the debuff, and Ironbark reduces damage and does good damage here.

I had about a 60-70% success rate with the DPH, but the switch to Crazy Carrot took almost any randomness out of it.

Carrot is definitely BiS for that battle and I've been trying to squeeze the best out of it! :)
A new strat is basically completed and I nag the environment as often as possible. I hope the result comes soon. If you can't wait, see the EU pet battle forums! :)

Another tiny variation of the same: Dark Phoenix Hatchling (1 2 1), Hatespark the Tiny (2 1 1), Fel Flame (H/P, 1 1 1). Start with Immolate, then hit with whatever highest hitting ability you have available. Make sure to keep target(s) Burning. All pets have similar abilities, so any pet can just carry on regardless of swaps.

This strat has failed on me 5 times in a row now. If you can't take out M-37 fast enough, you'll end up with too many stacks of his DoT and never survive to the end. Using Hatespark the Tiny in the first slot did work for me. He takes less damage.

I use a strategy that's similar but not quite the same:
Fel Flame - Scorched Earth - Is swapped
Pandaren Fire Spirit (Or any pet with Conflag) swaps in - Conflagrate
Fel Flame is swapped in - Conflagrate
Pandaren Fire Spirit - Burn - Enemy pet usually dies
Burn
Burn
Conflag - Pandaren Fire Spirit likely dies.
Fel Flame - Conflag, Scorched Earth - Burn
Ragnaros - Or any pet with Conflag&Burn or similar finishes the last pet off easily

Thank you for this, it was the only strat that worked for me

This needs more upvotes/to be a strat available at top. Works really well/reliably

Had some more tries and the strategy here is iffy at best. If your burn deals not enough damage, if you get crit, it's game over.

It's worth noting that, for the first pet slot, I tried to use Harbinger of Flame first and it didn't work, then I used Hatespark the Tiny and it worked perfectly, so having an elemental that takes less damage definitely makes a difference.

Yeah, any 3 pets with the Burn / Immolate / Conflagrate combo will work, provided the first two slots have extra 242+ speed as stated. I used a H/H Fel Flame, a Pandaren Fire Spirit and a Phoenix Hatchling.

I am using phoenix hatchling pandaren fire spirit and lil rag I mean its what I had and it works if I don't get unlucky crit