Tried it a few times before I left for work, and it does work as I outlined it. *edited to add a few thoughts*
I also remembered that there was another reason I wanted to make sure the Sapphire Cub didn't do to much damage. If it all went according to plan when Razortooth rotates in 2nd bot has about 960 hp left. When Razortooth finishes him he leaves the 2nd battle with near full health. As long as you have over about 1250 HP for the 3rd bot, barring any unfortunate RNG problems, Razortooth will have time for the second rip that I added
You might be wondering why 335 hp for the Sapphire Cup is the magic number? He will need to take 1 Metal Fist attack for plan A to work and it can crit for 334 HP of damage, so he needs to survive that. If you are feeling lucky and want to run the risk that Bot 2 doesn't crit that Metal Fist, you need over 223 HP, first time I lost cub to a crit I never again took the risk, I always go to plan B if Cub is under 335. Beside plan B is a faster finish by 1 round.
Durring Plan B if Cub Crits his pounce it can cause problems for Razortooth, which is why I really prefer plan A because a crit on screech won't do to much extra damage. You absolutely have to get screech on Bot 2 to make him slower than Razortooth.
If the carry pet is around 9 or higher you could leave out that second rip at round 13, this would be a similar finish to what Jerebear originally posted. Basically the ion cannon will go off before Bot 3 goes down for the first time and activates Fail-safe (mech revive), which is why the pet will need some levels to do enough damage before the cooldown on Ion cannon is over. With the second rip in place the bleed damage alone will kill him so the carry pet can be level 1 and never need to hit.
I personally leave the 2nd rip in every battle regardless because if I don't my flow gets off and I screw something up.