I know this is kind of late for me to be replying.
I have mega mega allergies (to the point where I've had multiple surgeries to open up my nasal passages and airways more), but I'm more concerned about my chinchillas having the best quality of everything that I can give to them. If I'm worried about bothering my respiratory system (which is almost always, but it's worse during certian times of the year) I wear a surgical mask when I'm around the dust, when I'm cleaning the dust, when I'm replacing the dust. They prevent contaminants from getting into my airways and they're avalibale all over the place for super cheap.
I like blue cloud because it's easy on my allergies and its one of the best dusts that I've tried thus far. Out of all the baths I've tried, the superpet one that you have (pentagon) kicks out the least dust. I do not offer the dust to my chins in their cage because it just makes a mess in the cage making it harder to clean and too much of it floats into the air during bath time. Instead I've fashioned the dust bath sideways into the shelf of a cheap bookcase that provides a "cover" (the shelf above) and the side opposite the door also works to catch dust instead of allowing it to be released into the air. I wipe down the dust bath area once a week to prevent build up of dust and when it is not in use I cover the hole with a piece of plastic wrap which I hold in place with an elastic band.
For hay I'm lucky enough to not have skin allergies (you didn't specify but I'm assuming based on how you're speaking that you have respiratory allergies) but again, I get the best hay I possibly can. I can't remember the brand of it off of the top of my head but I know that it's prarie hay and it's fresh and my boys love it and it doesn't produce much dust. I keep all of the hay, shavings, and extra dust in resealable containers (rubbermaids that I bought at Wal-Mart) that keep all contaminants out of the air and also assist in keeping the freshness in.
I know it seems like a lot of extra effort, but really, you don't have to deal with allergies as much, your chinchillas get the best care possible. So the effort is way worth turning a crap situation into a definite win-win situation, yes?