I would say contact Disney via their guest services to help you. But I do not think they are keen on their characters reading things that are not from their own universe and in all honesty I'm sorry, but I don't see it happening what you have in mind. Although I must say it is romantic
During Belle's Enchanted Tales is not really possible, especially because the room is filled with anxious children nervous to play their parts and anxious parents to film them. They might not all appreciate you stealing the thunder from their child. Also doing it at the end would not be ideal, as you would have to rush. The rest of the group cannot leave the show till the show is over and Belle has disappeared. Unless... you get the attendants to be in on it, and arrange that you could stay after everyone leaves and Belle comes back, but then reading River Songs Diary is still something I don't think will happen.
Other options are France or Akershus, but I think France is a better option, and you have more room to take pictures and it has a better backdrop. Plus not surrounded by dozens of nosy guests in the restaurant. It's also a little out of the way, so if your girl gets self-conscious on being the center of attention, she will be less noticed when you do it in France.
My suggestion would be: Lose your girl for a few minutes (say you have to go to bathroom, browse some shops whatever) at the end of one of Belle's sets. Before Belle leaves, go to the character attendant and let him/her & photopass in on your plan and decide on a return time. Return to your girl, suggest to see Belle at X time. What you need is Belle to distract your girl with questions about romance, romantic stories and princes (Belle must have something in her repertoire), your girl will need to have her back to you. You get on one knee and I'm sure at some point Belle will turn your girl at the right moment to face the other way.
If you want to use the Diary in your proposal, you best read from it yourself.
Good luck and let us know what you did and how it went!