1) it has to be effect damage, not battle damage.
2) it has to happen all in one round. So no burning 500lp per round.
There are 3 simple ways to complete this.
1) use Gravekeeper Vassal
2) use Destiny Hero Plasma when Crowler has 4000+LP
3) use Elemental Hero Flame Wingman and destroy an Ancient Gear Beast. This is the one Konami want you to use, since Flame Wingman is also part of the missions
It's not damage in the game mechanic sense. It's either an LP loss (similar to The Immortal of Thunder) or just changing the value to half the current value (similar to Unified Front for ATK). Which of these it is is irrelevant (no card actually checks for either), but it's not damage. (And would say it was if it was.) Whether the mission counts it or not, I don't know, however.
This is the Deck I used and Completed this 2k Effect Damage to Dr. Crowler's Ancient Gear Beast (Unlock Mission): Have to Fusion Summon Wingman and make sure that the Ancient Gear Beast is in Set Mode. Easy!
With SK you change AG Beast into def, Blazeman is for Poly search, Wingeg K is for stall, Goddess for Fusion sub, Ladybug for stall, Enemy controller for stall(or AG Beast to def), Block attack is for AG Beast into def.