The three Fly Swatter achievements are cumulative both through the Gaea campaign and the strategy game. To destroy a Manta from direct control, you must be driving a Walrus, flying a Manta, or using the turrets on the carrier and shoot down a Manta controlled by the enemy AI. Any Mantas that are shot down by your carrier, turrets, Walruses or Mantas that you are not directly controlling do not count for these achievements.
I would suggest doing this in the campaign, during the chase after Dr. Mao Shin. There will be an auto save created entitled "DeadLock - Volcano Combat" that is ideal for getting 6 kills in less than 10 seconds. You are in a prototype Manta with an incredibly powerful gun, so chewing through a few Mantas is a cinch. Rinse and repeat until you have 1000 kills, at which point the achievement will unlock.
If you choose to do this early in the campaign, the first point at which you face enemy Mantas and can shoot them down is Frontier, the fifth island.