This effect does not specify "and sends it to the GY". It triggers at the end of the Damage Step, even if destroyed after Damage Calculation (either by battle or by effect), as long as it destroyed a monster by battle.
The Ⓜ in this card's text is a maintenance cost. It is not an effect, does not activate, and can't be negated.
This effect targets 1 "Koa'ki Meiru" card in your GY.
If this card battles a monster with the same ATK and both are destroyed, its effect will activate.