Recipes I've seen for this tend to use baking powder, not baking soda, even though you're supposed to be able to use baking soda plus acid in place of baking powder. Perhaps that's more fragile than using baking powder?
Breads that use baking powder as leavening instead of yeast are sometimes called quickbreads. I've owned a few bread machines over the years and the recipe books that came with them all had quickbread sections, so the first place to check for recipes that work for your machine would be there.
Here is a bread-machine no-yeast bread recipe I found but haven't tried. The rise comes from a combination of baking powder and vinegar. I note that it calls for 2 teaspoons of baking powder and 1.5 tablespoons of vinegar, a ratio of approximately 2:1 acid to powder, while you tried a smaller amount and 1:1. If you can't switch to baking powder, I would start by doubling the baking soda and quadrupling the acid you're using.
With these breads you get something closer to batter than a dough ball and there are no long rise times (no yeast), hence the "quick" in the name. The breads range from cake-like to denser than regular bread -- fine for accompanying a meal, not so great for making sandwiches. They rise, but they aren't as light and fluffy as yeast breads in my experience. But your friend with the yeast allergy should still be able to enjoy them.