I had a one week old chick doing the same thing...gasping, and labored breathing with neck stretched. No other symptoms besides an occasional sneeze. No discharge from nose or eyes.
At first we thought it was something in her throat and tried the olive oil. After 2 days of gasping with neck stretched and very little eating and drinking, she started coughing (at least thats what we think it was). SHe slept with her mouth open for a few minutes at a time, but usually woke up gasping and sneezing again. We really thought she wouldn't make it through the night.
I started giving her a drop of polyvisol (no iron) mixed with 2 or 3 drops of water, hardboiled egg yolks mixed with yogurt, her starter crumbles, and a bit of warm water, into a kind or slurry several times a day. (remember, she was only a week old)
SHe slowly started improving and after a week of touch and go, she was eating, drinking running around like the other chicks. I am still baffled by what she had and that she recovered so completely. I researched and researched chicken illnesses and she just didn't have the other symptoms of anything I could find. My best guess is some kind of bronchitis or pnumonia. None of the other 11 chicks had any of the symptoms, so I doubt it was anything contagious.
My point in telling you all this is sometimes all they need is a little extra attention and care, which you seem to be doing a great job of. Don't give up hope for her. We were contemplating culling just as I noticed a slight improvement in her and I am so glad we waited. Your hen sounds very similar to my chick.
Good luck and I hope your hen recovers quickly.