Welcome to planDisney, and congratulations on your upcoming proposal! It's such a sweet way to plan such a magical moment.
While there isn't a way to book a fully private, one-on-one experience with Winnie the Pooh at Walt Disney World, you can set yourselves up for a cozy, memorable meet that feels personal. You'll find Pooh in a few places: at 
Magic Kingdom, he appears with Tigger at 
The Thotful Spot just outside 
The Many Adventures of Winnie the Pooh. At 
The Crystal Palace, also in Magic Kingdom, Pooh and friends visit tables from breakfast through dinner during the character dining experience. Both are adorable options, but I've got one more that I think will be perfect.
My top pick for the quietest feel is 
EPCOT's United Kingdom Pavilion, where 
Pooh greets in Christopher Robin's bedroom. It's an indoor setting with a storybook vibe, and it tends to feel the most intimate of the three. Perfect for popping the question if you time it right. 
I've got a few proposal-pro tips for you! Check the 
My Disney Experience app on the day of for exact meet times, and aim for one of the first posted sets, or one with at least a few more options after it. Lines can be cut if they get long, so earlier is safer. When you arrive, discreetly let the Character Attendant know you're proposing. They can't promise anything special, but they're wonderful at directing the flow of special moments. You'll definitely want to have 
Memory Maker so you can line up 
PhotoPass for the big moment. The photographer will capture the whole interaction, and they're very experienced at getting a lot of angles of big moments happening. Finally, if you choose Crystal Palace, let your server know the plan and ask for a heads up on character timing so you're ready when Pooh arrives at your table.
No matter which spot you choose, take a deep breath, savor the moment, and then pop the question. I'm already cheering for your happily ever after!
Have a magical day!
Jessica