Baking soda often is derived from corn. If you are highly sensitive or very allergic to corn or have sulfite sensitivity, then that may explain your reaction to baking soda.
An alternative is diluted food grade hydrogen peroxide (stronger than the hydrogen peroxide you get in the drug store). I don't know the proportions, but my Mom uses it instead of toothpaste or baking soda.
Another alternative is plain chicle - the original chewing gum base. It is good for the teeth & gums. You can get unflavored chicle from http://www.gleegum.com/refill-kit-ingredients.htm (I'm a satisfied customer with no financial connections with the company)
Chronic Lyme Disease, Gluten & Sulfite Sensitivity, Many Food/Inhalant/Medication/Chemical Allergies & Intolerances, Asthma, Gut issues (dysmotility, non-specific inflammation), UCTD ("Secondary Lupus-Like Syndrome"), Osteoporosis, chronic Lymphopenia, intermittant pancytopenia, chronic malabsorption/malnutrition, etc.; G-Tube; Currently TPN-dependent.
Meds: Pulmicort, IV Ceftazidime, Heparin (to flush PICC line), Claritin, Domperidone, Colloidal Silver (used topically), probiotics, homeopathy.