Unlike stainless steel jewelry, the stainless steel in the firearm probably won't have much direct constant contact with your skin. Most firearms have "grips" or at minimum "grip panels" that would be contacting with your skin. Another consideration is how long will the firearm be in your hands. Jewelry is likely to be worn for an extended period of time, so it could mean that you have direct constant contact with the metal for hours. Firearms OTOH will spend the bulk of the time holstered or on the bench. It is very rare that you will be walking around with a firearm in your hands for an extended period of time.
With that said, it's probably unlikely that your allergies will impact your ability to own or operate stainless steel firearms. There are other options for "shiny" metal firearms. A couple of choices would be chrome finish or titanium frame firearms.