My white coat is now grey! Help!

Sounds like you've tried the home remedies. Maybe it's time to take it to a professional dry cleaner.

Hey Cate - here is the answer from Patagonia on dry cleaning, this was taken from the Care Guide posted on patagonia.com and ifixit.com

Dry Cleaning

Given the rumpled nature of the road trips that inspire so much of our gear, Patagonia doesn’t make anything that requires dry cleaning. Our clothes are made to be worn and washed with very little fuss. More importantly, the EPA estimates that 85% of the dry cleaners in America use perchloroethylene, or "perc," to clean garments and textile products. This chemical solvent has significant human and environmental risks. We make clothes that wear and perform beautifully without all that.

Not sure if this solution is applicable as I first read about it when a chef wanted to make his uniform as white as possible.

Find a way to put juice from lemons in the wash. That's how he did it, but read up. I also read he used actual lemons instead of lemon juice. Your materials may be too different from his for this solution to be effective.

His whites were what car dealers call "super white".

The problem with white cars is that they only look their whitest in the showroom.

Don't ever take a down jacket, or anything else for that matter, to a dry cleaners.

Nellie has the best answer

After you try the washing method Nellie mentioned you could put it in DIRECT SUN for PARTIAL drying. Then, because it is down, I would tumble on low for the remainder of the drying time to thoroughly dry and refresh the down's loftiness (whatever the care label recommends is best). Sunlight does wonders on whites, especially yellowing armpits of t-shirts. Another trick for general whites is pre-treating stains with a paste of hydrogen peroxide, baking soda, and a tiny bit of blue Dawn worked into the stain with an old cleaning toothbrush (peroxide being the key "bleaching" ingredient, soda to activate chemical reaction and gently abrade fibers to loosen the stain, and blue Dawn as a degreaser).