Can walls be built on top of backer board?
You are trying to avoid a few backer board cuts related to the closet area. However, you are also increasing the height of the floor. If your existing walls are the standard 8' height, you will have to cut every stud of the new walls in order to frame them correctly. You are just creating more work for yourself. Either way you go, there will be an extra step involved and no significant time would be saved. Frame the walls first, then lay the flooring.