Why is the Golden Tooth so important with Robb having to bypass it through a hidden pass, Edmure holding the pass and Daemon Blackfyre having to break through the castle? Tywin didn’t leave through the pass, according to this quote ” All the time they were battling in the pass, Lord Tywin was bringing a second Lannister army around from the south. It’s said to be even larger than Jaime’s host”. Yet he can’t have been too far south as he crossed the Red Fork and didn’t go into the Reach.

The Golden Tooth guards the main pass between the Riverlands and the Westerlands, through which passes the River Road that stretches from Lannisport and Casterly Rock to Riverrun and Lord Harroway’s Town. So it’s a big deal because it’s guarding the direct route. 

My theory about what Tywin did is that he took the southern pass through Deep Den along the Gold Road, then hooked up sharply to attack the Mummer’s Ford. 

After all, Robb says that “Lord Derik had no sooner crossed the Red Fork than the Lannisters fell upon him, the king’s banner be damned, and Gregor Clegane took them in the rear as they tried to pull back across the Mummer’s Ford.” In other words, Gregor Clegane was on the east side of the ford before the battle started, and Gregor is definitely with Tywin’s army at all points after this battle, so it stands to reason he was with it before.