I don't think Sweets is all that bad, so long as you don't leave it in there for days on end.
The main reason doing that would be bad is because ammonia is ionic ... an electrolyte ... and copper is strongly cathodic, so if you leave ammonia in a copper-fouled barrel, you can get galvanic corrosion. Your barrel and the copper fouling in effect become a
battery with the copper supplying the positive charge and the barrel supplying the negative. But any electrolyte will do the same thing if two dissimilar metals are in contact with it, whether the electrolyte is an acid (like HCl toilet bowl cleaner) or a base (like ammonia) or an ionic salt like table salt (NaCl) or the potassium chloride (KCl) left by corrosive primers.
In general most metals are attacked far more by acids than by bases like ammonia -- aluminum and copper being two common exceptions. Bases (ammonia, lye, etc) are the opposite of acids, and they eat metals like copper and aluminum. But they shouldn't harm steel in the absence of copper or some other more-noble (cathodic) metal.
When I've got a lot of copper to remove, I skip the Sweet's and go straight to 10% janitorial ammonia, available at Ace/True Value hardware stores. It'll take your breath away, but it rips out copper fouling like you wouldn't believe.
).
Another another something with a lot of variables. 
Curious people have a hard time not asking. 
