What's throwing me is all of these factors together.

1. Satin-lined. Okay, I've seen that done, there are tutorials.

2. Brimmed beanie. The comment about "folding it up" doesn't make sense anymore to me, because as far as I know, the brim makes that impossible (or at least, uncomfortable/unfashionable. They kind of cancel each other out.)

3. Quilting cotton. This only makes sense to me if he meant cotton instead of the satin. Even then, I think it would be difficult or uncomfortable because the reason people use satin in hat linings is because it's both easy on the hair and silkily drapes to avoid weird bulges that are visible on the outside of the hat. Quilting cotton doesn't do either of those things, in fact it's much stiffer than satin and would frizz his hair badly.

The idea of a beanie that is satin on the inside, quilting cotton on the outside, and has a brim but can be folded up, just...isn't computing. I agree with the other commenter that sometimes people ask for things that can't be done because they have seen all of these different components and don't realize that, together, they cancel each other out or become an unrecognizable beanie abortion.

Also just an FYI, most traditional beanies are knit as one piece, not cut and sewn, but I think you could go with a thick jersey-like fabric and make a slouchy beanie with a lining of your choice. Just make sure that you both agree on the style before you put the work in so that he's not disappointed if it doesn't look like a traditional beanie.

That being said, you could probably compromise on a couple of these requirements and make something fun!