Ok, Was haveing extreme trouble pulling this off (influx of new players due to Humble Bundle) I can now do it rather easily, so I'll share how I went from "oh god!" to "this is a snap".
Shooting down the manta is nearly impossable without auto-aim. Seems this is one of the few games where you DO want that turned on.
Stay in standard optimal attack position: Above and behind. Not too close! You should not be getting hit by his attacks, only his allies.
You GENERALY ignore the additional enemies, but I pick off several as they get close to my line of fire. Stay on the main target for the most part, expecialy if his shield just went down.
That cannon can be fired almost non-stop. so I lay on that trigger pritty hard. I only give it the occasional brief second to cool as that is plenty.
You probably know his flight path by now, so you know where the hard turns are.
If his manuvers are a problem: You are probably too close and/or too low. Back off abit and stay well above your target.
If you are dieing to his attacks: You are WAY to close, and too low. Rethink your flight plan.
If you are dieing to his allies: That cannon can drop them in molments, feel free to strafe your crosshairs their way afew times. Right after his shield re-up is a great time.
Again: Turn on auto-aim!
I turned myself invinsable and ran my manta right up his backside (with autoaim off), he STILL managed to excape with 4% hull left. If he is excapeing your view area, you are doing something wrong.
Good luck Lt.