If you beat the game after getting the Hella Copter achievement, you'll unlock the Special Forces uniform.
And yeah, the Machine Gun will be one of the best ways to shoot it down. Stand on the rooftop (above the benches) then run toward it as it dives down to hit you - turn around as it flies overhead and aim directly up while shooting and try to hit its back rotor blade. You'll take it down before you know it.
Which set of benches?
The set closest to Paradise Plaza or the set closest to the Parking Garage!