I think the baking soda is used as an abrasive with the peroxide. I have a couple other ideas.
Try one of the shower sprays on a daily basis until the bottle is gone. Then after going through an entire bottle, if you still have the black color, try scrubbing it with an old toothbrush. I've used the shower sprays since the day the first one came out and swear it has saved me TONS of time and grief cleaning showers and tubs. When the shower glass and tile start to look filmy, I use a pot scrubber to wipe over and everything comes right off like magic. I believe the active ingredient is alcohol, you you might try getting a 90% denatured alcohol and spraying with that.
Certain things dissolve in alcohol that will not dissolve in water. Calcium is one of those things. Soap scum seems to be dissolved or eaten up by it, too. That's why the shower sprays work.
Or try Oxyclean. That stuff is a concentrated powder that generates its own hydrogen peroxide. Make a paste out of it with hot water and apply it to the mold in the crack. We used to have a neighbor who was giving us all her daughter's stained (but otherwise brand new) clothing for our daughter to wear. I cleaned them with Oxyclean, and after a couple months the neighbor asked me what I did to revitalize her daughter's old clothes. I told her and now we don't get any more clothes from her. It doesn't quite work like on TV but it does work. I soak clothes in a bucket overnight to clean them.
Too bad I don't get paid the Mr Oxyclean