You know… someone correct me if I’m wrong here, but in hindsight wasn’t Tywin taking a hell of a risk crossing the Ruby Ford and advancing north along the Kingsroad towards the Twins?
He’s got no idea that Littlefinger is keeping Lysa Arryn out of the war, and in fact he’s got to be wondering where the Vale levies are and what the situation there is. If she hadn’t already decided to hang her family out to dry and were simply biding her time or waiting until she had a full muster, she could REALLY have fucked him up; he’s got only part of his army, deep in enemy territory, with his right flank exposed. The Knights of the Vale could have smashed into him like a hammer, or even worse, seized the Ruby Ford behind him, trapping him in a box formed by the Mountains of the Moon, the Ruby Ford, the Green Fork, and Moat Cailin.
Advancing past the Ruby Ford seems dangerous unless you’re SURE the Vale will not bestir itself. I would go so far as to say foolhardy. And we know he doesn’t have scouts out, because Bolton steals a march on him and gets within a mile without the alarm being raised.
Although from a Doylist perspective, clearly the reason Tywin advanced to the north is because his army needed to be there for Tyrion to meet.
He is taking a risk, but you see Tywin as quite conscious of those risks in Tyrion VII – he stops at the Crossroads at a point where the road and the river means he can’t be flanked easily but can move his own men around quickly, and then creates some impressive defenses:
Well, there are earlier missives than Robb’s: Edmure sends a letter asking for help in Catelyn VII of AGOT, before the Lannister invasion of the Riverlands. Lysa’s refusal to help her brother, let alone Robb, is what gets Brynden Tully to quit as the Knight of the Bloody Gate.
So yeah, it’s possible.
Half a league from the crossroads, a barricade of sharpened stakes had been erected, manned by pikemen and archers. Behind the line, the camp spread out to the far distance. Thin fingers of smoke rose from hundreds of cookfires, mailed men sat under trees and honed their blades, and familiar banners fluttered from staffs thrust into the muddy ground.
…Three rows of stakes were pulled from the ground to make a hole in the line. Tyrion led his party through.
And then Tywin waits to see what will happen (and hopefully for Jaime to take Riverrun already and move up to support him), using Addam Marbrand as his eyes and ears. I at least got the sense that “unless the Starks and the Arryns come forth to oppose us” suggests that he was at least seeing the Arryns intervening as a possibility and had chosen that spot to counter-act that as best he could.
He only moves North when Robb Stark moves south, banking on his ability to beat Robb in the field before the Arryns can move, allowing him to refocus on Stannis.