Ok guys, I just completed this. This took quite a bit of grinding (save and quitting when I died), but I still finished at 19,900, which is well over the required 16,000.
I did this on Heroic with only the black eye skull activated. The open area parts are quite difficult when your shied doesn't automatically replenish, but the best option in these parts is to run past enemies rather than fight. Don't forget that finishing the mission quickly will yield a better multiplier. Keep the shotgun handy throughout for close quarters. Rack up the multikills with grenades and rambo-esque shotgunning on flood.
Of course, with black eye activated you can melee your way through precarious situations (especially those pesky outdoor bridge sections.) I recommend bashing flood as you traverse and killing everything you can with your shotgun (ammo availability permitting). I ran into a pickle with shotgun ammunition, but was able to make it work by keeping a few rockets handy towards the end.
Be patient and don't be discouraged by save-quitting and you should be able to make this happen! Good luck everyone!!!