Quatie
Free Ranging
This sounds like what happens when the food is powdery. It is more common with crumble feed with adults, and they can potentially inhale it. Pekins sometimes scarf down food more than other ducks, so I will see this with my girl sometimes.As for coughing- I don't hear it often, mostly if she is eating of kinda chokes. I'm not sure what the rattling would be.
You may need to put her in a bath to encourage her to clean properly. When I have treated my Pekin with bumblefoot, I would give her a bath when I would change the wrapping. She would come out of the bath and got the chance to clean and then new wrapping went on. I have found mine will not clean her eyes and nares unless she is in a bath. Yours may be the same.She's always had eye issues - in fact when we first got her she had pink eye, but one of her eyes had issues and a discharge a few weeks ago.
I would also recommend picking up some doxycycline. It sounds more like what happened to my runner. She stopped bathing when she went out and got a respiratory infection. I tried baytril, but it really didn't work all that well. Switched to doxy and it nipped it in the bud. Doxy is also more known for treating respiratory issues in other animals as well. You can still start with the baytril, but I would pick up the doxy as well. Baytril is great for treating bumblefoot.
https://allbirdproducts.com/products/doxycycline-20