What a horrible grind.
As near as I can tell, only "The Bloated Anomaly", "The Lichfield", and the lass boss count towards this challenge (feel free to correct me if I am wrong).
The Bloated Anomaly is by far the fastest and easiest of the 3. Unless you're going for medals on the other boss missions, I would advise doing this one over and over and bring a powerful wingman with you to speed it up even further, if you like. I also ended up getting bored and bringing some of my lesser dragons to give them some xp.
Edit: I've pretty much got these runs down to a science. Bring maxed Darkshoulder Tempest and a maxed wingman with a Flash Bomb nuke.
Phase 1: 50 seconds, shoot the balloons on his back.
Phase 2: As soon as it says "Entering Free Flight Zone" start spamming your nukes. He'll be dead before they're even done telling you about the free flight controls. Just over a minute per run.