Take a look at naval jelly, a weak form of jellied phosphoric acid sold under various brand names as a rust remover.
Several manufacturers say it is safe for removing rust from chrome.
Since it is a nearly clear gel, it is not supposed to leave any residue, commonly used to clean surfaces that will not be painted.
Usual caution here: test it on a small unseen spot before going crazy with this stuff. Also, note the instructions about how long to leave it on before wiping off (maybe 10 or 15 minutes, not overnight.)
Richard M