No, the poison's damage is not part of the attack and therefore not doubled
Death strike explictly states that the attack is doubled (emphasis mine):
. . . On a failed save, double the damage of your attack against the creature. (PHB, p. 97)
Damage from poison applied to a weapon has been clarified as being separate damage from the attack entirely, due to needing a saving throw, as you have referenced in your question. We can phrase this in this manner due to the ruling that poison damage is not doubled on a critical hit, and the definition of critical hit damage (emphasis mine):
When you score a critical hit, you get to roll extra dice for the attack’s damage against the target. Roll all of the attack’s damage dice twice and add them together. Then add any relevant modifiers as normal. To speed up play, you can roll all the damage dice at once.
For example, if you score a critical hit with a dagger, roll 2d4 for the damage, rather than 1d4, and then add your relevant ability modifier. If the attack involves other damage dice, such as from the rogue’s Sneak Attack feature, you roll those dice twice as well. (PHB, p. 196)
A couple clarifications here from the bolded text.
- The "attack's damage dice" are rolled twice. This means dice that are explicitly tied to the attack roll being a success.
- Sneak Attack is all-or-nothing and therefore part of the attack. If the attack misses, Sneak Attack does not occur. This is the same reason that Divine Smite damage is doubled on a critical hit.
- Poison damage is not tied to the attack, as the poison damage is not wholly dependent on the attack landing. The success of the poison is instead tied to a saving throw. The attack only matters in that it is the vehicle for delivering the poison; the rest of the poison damage mechanic is self-contained in the poison itself and not the attack.