I’m a little bit unclear about your scenario – the Ruby Ford is all the way at the southern confluence of the Trident, Tywin’s army is between Robb and the Ford, so that wouldn’t quite work.
To try something like that, Robb would have had to cross the Green Fork, go all the way down the length of the Green Fork, and then cross the Ruby Ford to the eastern bank of the Green Fork, and then all the way up the Kingsroad before Tywin deals with Roose. I don’t think the timing would quite work. If there were a handy flotilla of riverboats as at Tumbler’s Falls, maybe he could have boated down to get at Tywin’s rear, but it would have been very hard to avoid being spotted.
If I were Robb, I would have gone for something a bit simpler. Basically, do exactly what Roose did but do it right: march my infantry down to the hills, let them rest, then do a night hit-and-run with my cavalry to scare the hell out of the Lannisters and get some initial hits in.
Then I would have let Tywin attack my position along the top of the hills and use a combination of spearmen and archers to mow down anyone who tries charging up that hill, while keeping my cavalry hidden behind the line of hills to be used only in emergencies (for example, if Marbrand’s right threatens to break my position).
And when his infantry has been shot to pieces and his knights are lathered and de-horsed, that’s when I send my cavalry around his right, roll up the flank, charge my infantry down the hill and crack him wide open, and push his whole goddamned army into the Green Fork to drown.