How can low-carb bread rise?
If you're worried about feeding the yeast, you'll probably want to add a little sugar to your recipe. If you keep the amount low enough (I'd probably start with half as much as would usually be used for a given amount of yeast) the yeast will consume all or nearly all of it.

As to cake-like texture, that's as much a factor of lack of gluten development as it is the leavening agent. If you don't have much gluten going, you get that crumbly cake or muffin texture--which is what you want in cakes or muffins.

If you're going to use very little wheat flour or none at all, you might consider adding vital wheat gluten to your recipe so that you can build a decent structure in your bread.

I imagine you'd have to make hundreds of loaves of bread to get the balances right on your own, so I'd recommend looking for low-carb yeast bread recipes that contain a bit of sugar and maybe vital wheat gluten. Those would probably have a decent chance of success.