The wall accessory creates an object that is attached and associated to the wall geometry.
If that object does not allow you to align the textures then there won't be much that you can do.
But the question I have for you is why do you need the wall accessory for the image you have shown?
Just change the surface material of the external face of the wall to the one you want.
Then you can use the Align in 3D option to adjust the texture.
And if you do not want the surface to extend over the entire face of the wall, you can create a complex profile wall that has a thin skin for the texture you want to apply to it.
You can even create complex profile modifiers to adjust the height (top & bottom) and thickness of that skin so you can adjust it on each individual wall.