I addressed this in Cat VII of AGOT, but basically he should have done the opposite of what he did – rather than trying to defend every square inch of Riverlands territory, he should have relied on the rivers as defensive multipliers and in general, traded space for time while he was still mobilizing. 

Vance and Piper’s 4,000 men were never going to be able to stop Jaime’s army in the open field – but they could have shadowed Jaime’s advance and slowed it down, giving Edmure time to rally and drill his troops, and then gotten across the river ahead of Jaime. With these 4,000 men still in hand, Edmure would have been at least numerically equal and most likely numerically superior to Jaime’s army – but he likely still would have been outnumbered by both Jaime and Tywin’s armies, so I would not have advised going on the offensive at this point. 

However, as we saw in the OTL Battle of the Fords, the rivers give significant defensive advantages – they funnel the enemy into a limited number of crossing points, negating the advantage of the Lannisters’ bigger armies and allowing Edmure to establish local superiority of numbers, and the crossing itself leaves their forces exposed to missile fire and in danger of being wiped out when counter-attacked. The Tully river-navy could have been used here to great effect, both as siege platforms and essentially mini-castles that you can use to block crossing points. 

Essentially, Edmure should have done the Battle of the Fords in reverse. This may well have meant a siege of Riverrun – so be it, that’s what Riverrun was for. Potentially, he could have stopped the Lannisters there, allowing Robb to link up with the Riverlander army and fight a weakened Lannister opponent. But at the very least, it would have prevented the Riverlanders from being routed and allowing the Lannisters to run rampant with no one there to stop them. 

Even if Jaime and/or Tywin had managed to force a crossing, you leave a garrison at Riverrun to hold the castle, and pull back to the next defensive line – the Blue Fork (and then the Green Fork) in the north and/or various castles in the hills country in the south (ultimately, you’re going to be using Harrenhal as the place to stop any southern thrust). Keep making the Lannisters charge headlong at strong defenses, use your cavalry to attack the Lannisters guerrilla-style along the way just like the Riverlanders did in the Dance of the Dragons when the Lannisters tried marching through the Riverlands, and again, buy yourself time.