Reason neck size is important: military measures body fat by measuring around the neck at the adam's apple and at the wasit around the navel. Crappy way of doing it, because somebody with three chins can have a 45" waist and get away with it.
Bigger neck is easy...lots of heavy shrugs, medium to wide grip upright rows and good heavy deadlifts with good form. It may not be enough in 3-4 weeks, but if you drop your waist measurement and show some dramatic improvement, your 1st Sgt and Commander should take that into consideration and maybe grant your request for another weigh-in/measurement in 45-90 days.
Good luck.