A definite and quick death in a self-inflicted gunshot only happens if a very small specific region in the brain is obliterated. This region is only about 2.5% of brain weight and less than 3 inches long and has a width that ranges from 1/3rd of an inch to 1 inch.
To be able to aim so precisely is a matter of chance. In times of tense situations, it is harder to focus properly, hence the bullet can easily miss this narrow region. The suicidal person will be left with life altering injuries if death doesn't happen.
If you have doubts about shooting through the mouth then your doubts about shooting behind the ear are equally valid. One must always consider the consequences of a failed attempt, especially when such a violent method is used. People have been known to survive severe traumatic brain injury. There was an American named Phineas Gage who lived for 12 years even after a large iron rod went through his brain. Consider your options wisely.