The choice of baking powder for cake is based on two things. One is taste, not wanting a yeasty tasting cake. More importantly, cake flour and bread flour are different. Bread flour is high protein and high gluten flour. The gluten matrix forms during kneading and is elastic and therefore holds the carbon dioxide created by the yeast beasts and holds it for a long time. (Obviously, since you can let bread dough rise for a long time.)
All-purpose flour is in the middle, but closer to bread flour. It works for cakes. You're just not utilizing its gluten, and the cake is not a light.
Cake flour is low protein and low gluten, because you want a light cake, not a chewy bread-like product. And since no gluten matrix can form to trap CO2, a chemical leavening, baking powder, is used. So cake dough can't hold CO2 bubbles for long AND you don't want yeast flavor, so baking powder is ideal for short-lived dough. And you also have "quick breads" made with baking powder, for the same reason. Cake is just quick bread.
Why not just baking soda? Baking soda needs an acid to activate it, and that's usually something like lemon juice, but the reaction is very rapid, too rapid, and the gas can be lost before baking really gets underway. Baking powder is baking soda with one or more acid salts, and the reaction is activated by adding heat, which is when you want it to happen in a cake.
So if you're going to try yeast in a cake, it's a little tricky. You're going to get the leavening benefits of the yeast primarily during the first part of the baking, during that period before the yeast is killed. I suspect that will be hard to control. If everything works, you end up with a yeasty cake with peculiar crumb, since you will lose the great benefit of double-acting baking powder, double-acting because it has both low-heat and high-heat activated acids, so it can work over the whole baking period. Yeast can't do that. Plan accordingly.