The story from the perspective of Saruman would be a good idea.
Other interesting perspectives to tell the story from would be:
The Witch King (Not just a puppet of his master, though mostly so, and a failing puppet, at that)
Sauron (The story would start with the moment he becomes aware of Frodo and the Ring at the cracks of Doom, and be a flashback thereafter as to how he could possibly have got the whole thing so wrong.)
Elbereth (Trying hard to steer this whole thing to a happy ending from afar, through Gandalf, and Saruman, and Radagast, and the Blue Wizards, and timely dreams sent to Frodo and others, but it keeps going wrong. "What use are we Valar anyway?" Eru Illuvatar has to save the day in the end.)
Tom Bombadil (a passing, short, and fairly inconsequential incident in his much longer story).