Yes, the mucus itself could cause bad breath and when this mucus is ingested and comes inbto intestine, bacteria break it down and form gases.
ENT should know something about post nasal drip, sinusitis and chronic tonsilitis.
Gastroenterologist can deal with H. pylori infection of the stomach, small intestinal bacterial overgrowth.
If you have sulphur-smelled breath you should avoid foods containing sulhur.
http://www.medhelp.org/user_journals/show/16645?personal_page_id=801