The fact that the script differs from the book is also irrelevant
It's not entirely irrelevant -- depending on how a licensing deal goes, one could obtain the license to use material from the book, from the film, or both. If one is only allowed to use material from the movie or the book, one cannot use material from the other sources. This was often the case for the recent Lord of the Rings-licensed games, many of which were licensed to use characters from (and only from) the films, and were prohibited from drawing from the original books.
The fact that they differ might also mean that the IP holders value one set of material more than the other, which may impact negotiation depending on what the OP actually wants to negotiate for.
So,it is not a good idea to ask permission for a game featuring parts of the books. Instead, it is better to ask permission to use Tom, Jenny and Alice, in a new setting, and a new story.
Whether or not that's "better" is hard to say, it's entirely up to the person who holds the rights. I, personally, would be less willing to grant a license that lets somebody use my characters without using the rest of the world I've built around those characters. It seriously hampers the creative control I have over my creations.
Could I get the license for free, as long as in the game is is stated that it is not official and includes a copyright notice about the owners ?
You could get the license for free. You could also get struck by lightning. You're attempting to enter into a negotiation; it's not a science, and much of the result depends on what both individual sides of the negotiation want and what they are willing to give.
