Baking powder is baking soda plus an acidic compound. If you're using baking soda for leavening you need the acid of the buttermilk (or plain milk and vinegar) to make it bubble and rise (or yoghurt is it's acidic enough, ditto sour cream, some are and some are not). If there's also baking powder, it's probably for insurance, but different compounds in baking powder work differently, some when they get wet, others when they're heated, and some work faster and slower within their categories. Most bakers don't know or care, so they just try stuff until it works.

Too much chemical leavening can leave a kind of metallic taste, and you probably don't need more.

So, the thing about the bran cereal is that it was far more accessible in the mid-20th C. than sacks of bran. But another advantage is that cereal is generally baked. I'm thinking that might make them easier to work with. I haven't done muffins with whole bran, but I have done a lot of whole wheat bread and other quick breads. One thing they always warn you is that the bran in the WW flour is sharp and will cut the gluten. For quickbreads, which structurally are cakes, you're trying not to develop gluten, but I imagine the sharp bran could also puncture bubbles, but I don't know that for a fact. But since you're using all that bran and trying not to develop gluten, another thing is that your muffin batter isn't going to be very stretchy. That can also lead to less volume.

With bran muffins, you want to handle the batter as little as possible--all the good recipes say so. Don't overmix. Meaning mix just enough to wet everything.

The two eggs are a huge difference. Huge. Eggs are fluffifiers.

I'm sorry. This is disjoint. I hope it says the right stuff, but my brain is saying it wants to go to sleep. Good luck!