Agreed. Gun oils are probably your least invasive, best bet. The next step is some sort of abrasive polish, starting as light as possible. Since you don't likely have polishing lab supplies like I do, a good starting point there is the stuff made to polish kitchen appliances. "Stainless Steel Magic" and the like. These take a little elbow grease, but the abrasives in the product are small and will leave a nice shine. If you need to go more aggressive you could move to the stuff made for car paints. In order from least aggressive to most would be:
Polish --> swirl removers --> scratch removers --> compound.
If said stuff leaves a haze, follow up with something less aggressive till your happy, then oil.
And note, if your chrome is not actually chrome but rather a chrome veneer, there is danger of chipping it off.