What makes muffins soft is starch and fat. You have no fat at all in these "muffins", and very little flour when compared to the vegetables and proteins.
Normal muffin proportions are 2:2:1:1 flour:liquid:egg:fat (per Ruhlmann). I guess you can add up to 2 parts filler (so as much vegetable as flour) before you get the result too terrible. In your mixture, there is a total of 140 g flour and ~700 g fillers (bran, beans, whey, carrots, bananas). This is never going to be muffin-like, you will have to forgo most of your additions. And the whole wheat flour is also making it harder. You either have to change to normal flour, or live with some hardness (it should be manageable if you change all the other factors).
The liquid provided by the yogurt is good if you switch to white flour, no need to change anything there. You may consider adding a bit if you stay with whole wheat, it "soaks" more liquid.
The egg whites are a problem too. First, egg whites make baked goods dry out. You shouldn't be using pure egg whites, they will make bad muffins. You need whole eggs. And second, the equivalent of 4 eggs (200 gram) is probably too much. 1-2 whole eggs would be normal for this amount of flour, 3 makes it pound-cake-like but still usable. Still, you may find out that, if you make all the other changes, upping the (whole) eggs to 4 doesn't make it too bad.
You also used too little sugar. Sugar is an important structural element in muffins. It retains moisture and prevents gluten buildup. It can be reduced if you don't want the taste or the calories, but each gram you take away also reduces the quality of the muffin, so you have to try out and see how much quality reduction you can tolerate.
You had absolutely no fat in the recipe. The standard ratio suggests 75 g. This is also a major factor in softness and moisture.
In short, you cannot take any muffin recipe, remove all the stuff which doesn't have the nutrients you don't want, fill it up with foodstuff you want, pour it into a muffin form and expect it to taste like a muffin. Baking is chemically very complicated, and most of the ingredients are not there for flavor, but for texture. Replacing them with random other ingredients will do nothing good.
You can either return your recipe towards a more traditional muffin, which will of course change its nutrition values back to "sweet baked good", or you can find a different form of combining the foodstuff you have chosen to eat. Mixing it and baking it is not going to work.