Another lifelong asthmatic here.
One thing I learned with the hand-fired Harman is to shake the grates with the ash door closed (always kept it open while shaking to watch for embers -- makes a bit of dust in house over the season). With the door shut there is hardly any.
Instead of shoveling the piles that happen on the sides & back of ash pan, you could use a vacuum -- that would virtually eliminate all dust. Unless, of course, you fall up the stairs while carrying a heaping ash pan.
About 8 years ago, a doctor put me on Flovent inhalers -- BEST thing that ever happened to me. I used to go thru 1 albuterol inhaler every 2 weeks!! And they say it's bad if you use 1 every 6 weeks!
I wouldn't be able to keep my current job without it -- waaaay too physical -- requires ALOT of air...