Backstabs under the effect of Batallion's Backup still deal a lethal amount of damage, as this video sets out to show:
Critical hits deal 300% of the base damage of the weapon ...
The Spy's melee weapons always and only crit on backstabs, boosting the damage of a backstab from 2x the victim's current health to 6x.
Backstabs deal damage equal to 600% of the target's current health and always deal critical hits, unless otherwise prevented by a Battalion's Backup. However, a Battalion's Backup will not cause a damage reduction.
(Backstabs deal a base damage of 200% of the target's current health, and triple that [600%] on a crit. If the crit is removed, it will go back down to 200% of the current health; still lethal.)
Battalion's Backup does prevent backstabs from landing a critical hit (600% current health), however it does not reduce the non-crit damage of the backstab which is still lethal (200% current health).
While under the effect of Batallion's Backup, this would simply reduce the damage from 600% to 200% of the player's current health. They could have 1,000,000 health and a non-critical backstab would still deal 2,000,000 damage; a lethal amount. (6,000,000 if it criticals; also lethal)
Whether you deal 2x a target's current health, or 6x a target's current health, they are both lethal amounts. Anything equal to or greater than 1x a target's current health will be lethal.
The Phlogistinator's 75% reduced damage buff does not protect against backstabs.
Once filled, it can be activated with the secondary fire command or taunting (default key 'G'), which performs a taunt that will refill health completely and enable a short crit boost. During this taunt, the wielder has 75% damage resistance (however, this does not protect against damage from Backstabs).
Backstabs deal 10% damage to spies using the Dead Ringer. 6x current health will instead be 0.6x current health. 2x current health (a non crit backstab such as one being affected by the Battalion's Backup) will instead deal 0.2x current health.
The Dead Ringer blocks 90% of the damage from the activating hit as well as during the first 6.5 seconds of the feign death Cloak. This allows the player to survive fully charged headshots by Snipers, backstabs from enemy Spies, and Taunt kills as long as the user has a sufficient amount of Health before the attack was made. Since backstabs have a base damage of 6 times the victim's current health, and the initial hit that activates the Dead Ringer does 10% damage, backstabs will deal 6/10 of the Spy's current health.