Special thanks to Ockstar at xbox360achievements.org for his useful tip. Credit goes to him for this method.
Original post: http://www.xbox360achievements.org/forum/showthread.php?p=48...
Here is an expanded explanation along with my own additions:
Step 1: Fly to the right fan geyser and queue up 2 squads of Razor Girls to attack it. Play Call of the Wild (mine was maxed out so a Hextadon was summoned) and leave it here.
Step 2: Queue up a Headsplitter and play the solo "Rock Block"
Step 3: By now enemy troops should be contesting the fan geyser, ignore them and team up with your newly built Headsplitter.
Step 4: Drive the vehicle left towards the enemy stage. When in range begin firing, remain on the vehicle until it is destroyed.
Step 5: Fly to the far right side of the stage and use "Bring It On Home"
Step 6: If the stage is not yet destroyed, queue up another Headsplitter and then fly back to repeat step 4.
Using this method while playing on Brutal difficulty, I only had to use one Headsplitter followed by the "Bring It On Home" solo to achieve victory.