Chromed items are the worst. I just noticed my Tesla license plate frame is getting ready to shed its coating already after only 1 year ugh. Thankfully it's just the accessory.
For damage to chrome in the past I've used chrome spray paint, spraying it onto folded paper or paperboard and then applying with a fine point (eg bamboo skewer) to fill. The chrome spray paint has some metallic / pearl to it, not exactly giving you the same chrome effect but giving you the general right shade.
Then again these days there might be a better fix, maybe Dr. Colorchip has something... I just used their touch up paint kit this morning on some new hood chips.