This is a good question, ultimately you can just leave the baking soda out, it will still be an effective deodorant. For consistency you could increase the arrowroot powder a little.
Otherwise, you could try diatomaceous earth (DE), which has some similar actions as baking soda but is less irritating. When purchasing the DE, make sure it is food grade. I would say you could leave the amount about the same as the baking soda was in the original recipe or increase it a bit (1 tbsp-1.25 tbsp).
If you try out the diotomaceous earth, I would be interested to hear how it turned out for you!
Happy product making,