I suggest that there are three main uses of a sword: cutting, thrusting, and blocking. With my limited knowledge of German longsword, I'm going to analyze the effectiveness of a mouth sword in all three of these tasks.
Blocking: When blocking, it's best to try to block off cutting lines. If you hold the sword out towards your opponent, you're not blocking anything; they can cut you from any angle just fine. If you hold the sword sideways (as in the picture), any cuts you deflect are just going to be redirected to your head or neck. The best bet I can think of is to have the sword facing slightly down and to the side, but that's going to get in the way of your primary sword (which could just block that line normally).
Cutting: You have to turn your head to cut. That means you'll lose eye contact with the enemy, which is a bad idea. You'll also probably have to move your upper body, which could expose your back.
Thrusting This is the only one that I think might work. If you have your third sword sticking out like a tongue, it could serve as a last defense against someone who would otherwise try to close the distance. However, by the time they get close enough to be stabbed by your mouth sword, they could've already stabbed you with a regular sword, or probably even a reasonably long knife. And, failing that, you've just given them a lever to break your neck with.
So no, I don't think a mouth blade would help that much. That said, maybe if you carried a third sword with your feet, you could pull off some effective maneuvers.