I was once going to duplicate one my MIL had for her. The original was made of wool and the "collar" was more of a scarf that was mostly attached, can't remember exactly. I never did make it because the original was a single layer of wool and the edges were all finished with a over lock type stitching and I don't have the that machine.

Late to this, but I'm surprised you did not find these pages I got when I used the keywords 'akutsaki cloak' on GoogleImage.

Looks like it's not really a cloak, but a coat w/a standup collar, worn over the shoulders like a cloak...?

Here is a link that might be useful: Akutsaki cloak @ GoogleImage