Thanks to Sqqueeky on wowhead, this strat worked the best for me if your pets atleast have 244+ in speed
Slot 1: Fel Flame 1-1-1
Slot 2: Phoenix Hatchling 1-2-2
Slot 3: Dark Phoenix Hatchling 1-2-1

Turn 1: Immolate
Turn 2: Conflagrate
Turn 3: Conflagrate - Corporal Hammer revived
Turn 4: Immolate - Corporal Hammer dead
Turn 5: Immolate
Turn 6: Burn
Turn 7: Conflagrate - M-37 revived and then dead
Turn 8: Immolate
Turn 9: Conflagrate
Turn 10: Burn - Clamp revived
Turn 11: Burn - Clamp dead

For me it worked best just to do as of these turns. No need for crits and I havent tried that many times yet. But it left my last pet to have 1k HP left at the end. You should end up with only 3 dots from M-37 but should work with 4 dots aswell.

If I messed up the last turns please tell me. but this should work good enough

Here is my script, It has been working perfectly the last 3 times and seems to be a bit faster.

It delay the killing Clamp and heal up Crazy Carrot to 1000 hp before starting the fight against Corporal Hammer. Maybe I'm on the safe side setting the limit to 1000 hp. You can try to lower the requirement by changing the line:

ability(Bleat:539) [ self(Crazy Carrot:1396).hp < 1000 ]
to
ability(Bleat:539) [ self(Crazy Carrot:1396).hp < 800 ]

Script start ====

if [ enemy(Corporal Hammer:2068).active ]
if [ self(Crazy Carrot:1396).active ]
ability(Bash:348)
ability(Aged Yolk:667) [ self.aura(Firebomb:1918).exists ]
ability(Ironbark:962)
endif
if [ self(Gleamhoof Fawn:1913).active ]
ability(Hoof:493) [ enemy.aura(Mechanical:244).exists ]
ability(Bleat:539) [ self(Crazy Carrot:1396).hp < 1000 ]
ability(Nature's Ward:574) [ !self.aura(Nature's Ward:820).exists ]
ability(Nature's Ward:574) [ self.aura(Nature's Ward:820).duration < 2 ]
ability(Hoof:493)
endif
endif

if [ enemy(M-37:2067).active ]
ability(Ironbark:962)
endif

if [ enemy(Clamp:2066).active ]
change(Gleamhoof Fawn:1913) [ self(#3).active ]
change(Gleamhoof Fawn:1913) [ enemy.round = 1 ]
if [ self(Gleamhoof Fawn:1913).active ]
change(#3) [ !enemy.ability(Storm Coil:1922).usable & !enemy.ability(Drill Charge:1921).usable & !enemy.ability(Clamp:1920).usable & !self(#3).played ]
ability(Nature's Ward:574) [ !self.aura(Nature's Ward:820).exists ]
ability(Nature's Ward:574) [ self.aura(Nature's Ward:820).duration < 2 ]
ability(Bleat:539)
ability(Hoof:493) [ !enemy.aura(Mechanical:244).exists ]
ability(Hoof:493) [ self(Crazy Carrot:1396).hp > 1000 ]
ability(Nature's Ward:574)
endif
endif

Haha, I also like to start beating Hammer with Carrot having at least 1000 HP!
I'm not pasting it yet, however, not because the text requires otherwise, but because the strat environment will undergo a maintenance in a couple of weeks. :)
Thank you very much.

Anyone brave enough to tackle this TDBattlePet script?

Sure, I'll give it a shot. I tried this one out a couple of times and seems to work, but might need tweaked. I can also post the longform version (with all the names) if wanted. Let me know how it goes.

if [ enemy(2068).active ]
use(962) [ enemy.round = 2 & enemy(2066).dead ]
use(493) [ enemy.hp <289 ]
use(348)
use(539)
use(574) [ self.aura(820).duration < 3 ]
endif
use(667) [ self.aura(1918).exists & enemy(2067).dead ]
change(1913) [ enemy(2066).active ]
use(962)
use(574) [ !self.aura(820).exists ]
change(#3) [ !self(#3).played & !enemy(2066).ability(1920).usable & !enemy.ability(1921).usable ]
use(539)
use(493)

Creator's note:

Yes, it is slow. Each battle with Bert is relatively slow. It is bothersome to follow. Moreover, it might be more successful when you adjust your reaction to what's exactly happening, or delay killing Clamp to get more AoE healing.
Still, this is a levelling strat. In a battle where you have a front pet DoT, AoE, forced swaps and a bit of uncertainty. As you learn it (or get a really good script), however, you'd be able to repeat it everyday not only just for the pet, but for your alt aswell.

And yes, I know about other strats with trainee pets. One uses treants, another one uses Molten Corgi and Hatespark the Tiny. :)

About the carry level: level 6 describes mostly the beasts. Most pets with the 4rd should suffice. Magic pets can be 1, and if you don't fear the crits – elemental, mechanical and a couple of H/H pets can be 1 aswell.

Worked well. I used a P/B Fawn and still finished on same turn as strategy posted first try. 4760xp on lvl11 /w safari hat to lvl 16.

Worked perfectly for me as well, very different, I enjoyed all the healing. I had to go get a P/P Gleamhoof Fawn, and I love it. Thanks!