To sum up my answer, no. It can't. If you want to see my full and rambling answer, read on.
Iron, as we know, reacts with oxygen over time to make iron oxide. Contrary to popular belief, this actually takes a hell of a lot of time if left alone; iron in the middle of the sahara would take hundreds of years to noticably rust. However, the process is sped up by water, and even more by halite (salt) and many acids.
Bronze, as you most likely have realised already, is an alloy of tin and copper. Tin can react with oxygen over time, creating a white powdery type stuff. Copper reacts very slowly indeed with oxygen, barely at all. Instead, it reacts more quickly with carbon dioxide, to create copper carbonate, which is the green colour, or it can react with salt to create copper chlorite (green again) or with sulphur to create copper sulphate (black tarnish. very weird looking). Now, the chlorine and sulphur will react much quicker than the carbon dioxide with this.
In a shallow marine environment, you'll find very little sulphur, so this is effectively null. On the other hand, there's plenty of salt, and the shallow depth means the water will be rich in oxygen and carbon dioxide compared to a deeper marine environment. Assuming temperate temperatures (western europe, for example) and a normal level of salt in the sea (atlantic ocean) we're probably looking at pretty quick reactions here. Unless the mass of bronze is very large it won't last more than a few hundred years, and within a few decades it will be practically unrecognisable. A sword, for example, would probably become unusable within ten years.
Another factor to consider would be that shallow sea water is usually a high energy environment; part of why it would react so quickly is that it would be continually being swept about by currents, waves and living creatures, which would scrape off the reacted surface and allow easier access to the pure metal beneath.
You could slow these processes by lowering the temperature or salt content. Very still water would also help by lowing movement, oxygen and carbon dioxide. Really though, you want to bury it in fine mud or silt which will cut it off completely from the various reactants (as the helmets you mention were). To be honest, if it's in a shallow marine environment for two thousand years, burial is inevitable anyway; these areas have a large level of deposition.