Don't use a microfiber cloth, use an edgeless cotton cloth
While many resources talk about using a microfiber cloth to clean leather, depending on how hard you rub, the fibers on a microfiber cloth can change the texture of a leather bag, especially if it's pebbled.
Microfiber cloths are usually made out of synthetic fibers1 and many of them are made to clean glass and other smooth surfaces2. While smooth surfaces can be scratch prone, they are stronger than animal skin.
It is recommended to use a white cotton cloth instead. This is what Docride suggests, a specialist in restoring Hermes bags.

Microfiber cloth ❌
Cotton cloth ✅Use a micellar water for sensitive skin
Drip a bit of micellar water onto the cloth and rub gently on the stain. Sometimes, this can be all it takes to remove the stain.
Stain not coming off?
You can either choose to live with it or go to a handbag repair specialist who is able to mix a color matched dye and carefully apply it to cover the stain. As tempting as it may be to try to remove the stain yourself, while there's a chance it could work, the risk of further damaging the bag is usually not worth it.
It is not recommended to go to a cobbler that mostly fixes shoes because they might not have the expertise or tools to restore your bag as well as a handbag repair specialist could. They might not be able to mix a color matching dye or have an airbrush to apply it in thin layers.
Sources
1. https://kaivac.com/clean-academy/the-science-behind-microfiber-cleaning-cloths
2. https://www.tapplastics.com/product/plastics/cleaners_polishes/microfiber_cleaning_cloth/119