"If anyone has any suggestions (baking soda doesn't work either and it's way too abrasive to use for brushing teeth)"
I have been brushing my teeth with baking soda for close to 25 years. At my last dental check-up, I specifically asked my dentist if she thought my tooth enamel had degraded. She said it was great. My perioidontist said, "Whatever you are doing, don't stop. Your gums and teeth are fine."
Perhaps baking soda has a bad rap because it is often combined with salt, which is abrasive.
The following is from an Arm and Hammer web page:
"It is an old-fashioned misconception that Baking Soda is abrasive. In fact, baking soda is very low in abrasion. The abrasive properties of a dentifrice are measured in a standard test called the Relative Dentin Abrasivity (RDA) Test, which is conducted by accredited facilities. The abrasivity of the dentifrices are recorded as RDA values and used by all toothpaste manufacturers. The higher the RDA value the more abrasive the dentifrice. In this test, baking soda alone scores a 7, and the ARM & HAMMER® products ranging from 10-117, all within accepted levels of abrasivity and typically below most other toothpastes."
Albeit, this is from Arm and Hammer who in the baking soda business, but the condition of my teeth bear witness to their claims.
Good luck with cleaning your stains.