Personally, I'd rehome the bullied hen. She obviously is a more docile temperament. Is she a different breed than the other two?
If you do decide to put her down, with the idea that the others may be picking on her because they sense some health or neural imperfection (can happen), I prefer the cervical dislocation which is quick (if done right) vs the killing cone which honestly takes awhile to bleed out.
Get a broom, place hen face down on the ground while holding feet. Place broom handle at base of skull. Step on both sides of the handle. Quickly pull back and up on the feet. You should feel and see a pop, snap, at the neck. If done aggressively, be forewarned you can end up beheading the bird.
If you are at all squeamish, or want the absolute gentlest way out for the hen which is AVMA approved, I highly recommend the CO2 method. Get a 5 gal bucket with lid. You will need to procure some dry ice from your local grocery store, about 1/4 pound, which should only be a couple of bucks. (Be careful to only handle with gloves to prevent freezer burning your hands). You will need to use within a couple of hours as the dry ice will melt even in the home freezer.
Place dry ice at bottom of bucket. As a kindness place some Styrofoam or such packing over the dry ice so the bird doesn't sit on the ice, while being sure it is not a tight seal over the ice. Add warm water. Immediately a cloud will begin to form of CO2. Let this cloud build in the bucket for several minutes by lightly placing lid over bucket (but not tightly sealing or you risk blowing the lid off from pressure).
When a big cloud forms, lower bird into cloud of CO2 and replace lid to keep cloud within bucket (again being careful to not seal too tightly for possible pressure overload). The hen will take a quick gasp. Unconsciousness immediately follows. Death from asphyxiation occurs within about 30 seconds to 1 minute. The bird will have involuntary muscular flutter. Wait another minute after fluttering ends to assure death has occurred.
Good luck with your decision for this hen. Flock drama is always difficult.
LofMc