If there is obvious mist (steam) rising from the lye solution, you definitely don't want to breathe the mist. But that stops fairly quickly after the lye solution is fully dissolved. Keep the top of the lye solution container lightly covered to reduce the amount of mist that escapes into the open air. And work in an area with good air ventilation.
I don't choose to put my face near lye solution or soap batter, but if there is no mist in the air from these liquids, it's normally fairly safe to work nearby without a respirator. Are you mixing the soap batter in such a way that droplets of the batter are getting into the air? If so, you need to find another way to mix the batter that doesn't do this.
You haven't said anything about your recipe, so I have no way of knowing if your other ingredients are causing this problem.
What I can say is if you ever breathe lye mist, even a little bit of the mist, you will quickly learn it's not a smell. Your nose and possibly your eyes will burn badly. The irritation is not an odor and it's not something you can ignore.
Lye mist doesn't travel far from the source, so if other people farther away are smelling this, it's not the lye solution -- it's some other problem.