I'm not as expert on long products as on flat, but 303 with all its elongated sulfides may present some problems with splitting. Why not use a low sulfur grade which would also have better corrosion resistance?
In general you pick the stainless grade which resists the given environment and doesn't need coating.

Since most rivets are cold formed they tend to be made from 301 or 302.  Often they have coatings, but htat is very application dependent.

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Corrosion never sleeps, but it can be managed.
http://www.trenttube.com/Trent/tech_form.htm

I concur with McGuire’s comment about potential splitting of 303.  Also I am not sure there would be any reason to use free machining S.S. except for a machining application especially since EdStainless is pointing to the standard S.S. for rivets.

Passivation is one of the most common surface treatments for enhancing corrosion resistance of S.S. but  I would be most interested in hearing suggestions by others for your particular application.  Please elucidate with respect to environment.