The Route of the Ring

[The Might]

I also wonder why there is no mention of this option, especially since we're told the pass was safe due to Grimbeorn the Old and his men. So they would have anyway received help from Grimbeorn, who would perhaps know of Gandalf or Hobbits from the stories of his father. From there it's not necessary to cross the Anduin, but simply head south towards Lorien (I anyway think they would have done this intead of crossing, since the eastern shore was guarded).
The only argument against it is that it might take too long, but in fact I think it would have been a better decision.