Well, I think this is a case of “"any Coburg prince.” I.E, I don’t think you necessarily need a Robb Stark – Robett or Galbart Glover, the Greatjon, really any competent commander could have done a much better job.
As I write about in my recaps of Tyrion VII and Tyrion VIII of AGOT, I think Roose Bolton deliberately botched an eminently winnable battle. To begin with, deciding to go on the offensive was odd, given the Northmen were outnumbered and lacking in cavalry. His night march is especially weird, because A. he knows that Robb wants Tywin’s army as far away from the Trident as possible, and B. despite achieving tactical surprise, Roose gives the game away by drawing up in battle formation rather than attacking Tywin’s army immediately.
But even more egregiously, given that Roose has an army of almost entirely foot and a strong defensive position on the high ground that stood perpendicular to the Green Fork itself, his decision to send slow-moving spear-and-shield infantry to attack Tywin’s cavalry vanguard is absolutely insane. As in the Battle of the Hastings, if you put a shield wall on the heights of a long line of hills, knightly cavalry cannot break through – the uphill climb makes it all but impossible to generate the speed you need for a proper charge, horses founder and fall – wrecking formations. Likewise, the heights make an infantry assault even more difficult – it’s incredibly difficult to stay in formation and your shield up while you’re climbing, massed pike formations draw devastating plunging arrow volleys that gain momentum and range when you’re firing down-slope, and the Lannister bowmen’s shot is going to fall short.
What’s bizarre about the Battle of the Green Fork is that Roose Bolton deliberately does what Harold Godwinson absolutely sought to avoid throughout the whole battle, without the feigned retreat, the faked death of William, and the actual death of Harold that was necessary to get the Saxon housecarls to break discipline.
So with a competent and/or loyal commander in charge, the Northmen take the heights, form a strong shield wall to screen their archers, and hold their position, forcing the Lannisters to make repeated up-hill attacks that are bloodily and repeatedly repulsed. Once the word gets around that the Young Wolf isn’t present on the battlefield, and the Lannisters are forced to withdraw, the Northmen pursue relentlessly down the Kingsroad rather than retreating back to the Twins, increasing the Lannisters’ OTL losses from the forced-march retreat.
If at all possible, try to maintain contact with Tywin’s army and slow them down as much as possible, creating the possibility that Robb’s combined Northmen and Riverlander cavalry force can beat Tywin’s mixed force to the Trident, trapping Tywin’s army between the Green Fork and the Ruby Ford and the mountains, potentially knocking the Lannisters out of the war in one fell swoop. But short of that, make sure to seize the strong defensive position at the crossroads and take control of the Ruby Ford, which Roose doesn’t do until Arya VII of ACOK. This creates a strong defensive line anchored by the Red Fork of the Trident, ensuring that the reaving never spreads “north across the Trident almost to the Twins.” It also gives Robb a protected seaport at Saltpans which allows communication, reinforcements, and resupply from White Harbor.