No, dispel magic does not cause HP damage. Disenchanting something is not the same as killing something.
1. No damage is dealt
Your combo works off the following interaction.
The simulacrum spell description says:
The simulacrum lasts until it drops to 0 hit points.
The description of death ward says:
The first time the target would drop to 0 hit points as a result of taking damage, the target instead drops to 1 hit point, and the spell ends.
Dispel magic deals no damage. It simply removes the simulacrum spell from the target. Sequentially, the simulacrum would revert to snow / ice and melt instantly. Its HP never hit zero as a result of damage.
The same could be argued that you could disenchant a construct and it would stop moving. Any damage to the structure of the construct would remain persistent. If you were able to animate it again - it could be argued the re-animated construct would could have the same number of HP it had before being disenchanted unless time was taken to repair the construct in some way.
2. The simulacrum is never actually killed.
Another part of the death ward description says:
If the spell is still in effect when the target is subjected to an effect that would kill it instantaneously without dealing damage, that effect is instead negated against the target, and the spell ends.
The simulacrum is never actually killed. It is dispelled. Again reaching up to my construct example from before - the construct's form still persists without its enchantment, but it isn't outright killed. Neither dispel magic nor ending a spell is a specific death effect. It creates a death-like effect once it's disenchanted.
Additionally, summoned creatures (Summon Monster and its variants) that are dispelled aren't killed either, they are dispelled. And do not trigger necromancy like effects such as Dark One's Blessing (warlock) or Grim Harvest (Wizard).
Relevant Topic: What happens when you target dispel magic at a summoned creature?