The "fumes" are most likely to be tiny droplets of Ferric Chloride solution (with the copper chloride products, and MAYBE some HCl.) In industrial quantities, and with a spray machine, I'd be a bit worried as well.
OTOH, the particles should be relatively easy to filter (big "droplets" rather than tiny molecules.) Probably anything good against dust would work fine. (which alas has little to do with what the health and safety people might require.)
If it's really noticeable, I might worry about my eyes, too...