I suppose brushing does help the bad breath problem to a certain extent if you brush your tongue to get the sulpher muck off, but since the lungs are one avenue of elimination, you will still have BB to some extent. Also the Sinuses are going to be draining, and they can be a source of bad breath.
Oil pulling might be good as it eliminates mouth bacteria, and also really makes the mucous flow. Just avoid swallowing.
Also you could irrigate the nasal, and sinus cavities with warm distilled salt water with a little added H202 to kill bacteria. Again doing so over the sink to avoid swallowing anything.