"I think the film's take is more plausible, if there is not an automatic appointment to the throne, there was no immediate need for a single king, and there was disagreement over who should take the throne."
What we have Peep is a man who killed his brother, the previous king. The young prince, heir to the throne, is too young so the next in line, who happens to be the king's killer, is now the Lord Regent. I would believe it more that this man who wanted his brother's kingdom would make more subtle moves during his regency to build up his power over the years to make himself king. It happened in the book, but not in the movie.