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As the title suggests, the larger gaps are to be filled with Kumiko. The smaller gaps will be left empty. I though about a simple pattern as in the demo lantern on the right in this picture.
Do you have other ideas? All I can currently do with my mitre saw are square patterns. Cannot do triangles, unfortunately.
Awesome! Unfortunately, that's exactly the kind of triangle pattern my mitre saw is too inaccurate for.
Make yourself a shooting board at the right angle.
If I understand you correctly, that wouldn't help with the 30° cuts into the strips that make up the frame. Perhaps I need to upgrade my jig for that, but I haven't yet found a solid and easy approach to that.
https://youtu.be/JbpwDufvzSo
This is a 90 degree shooting board. You can make one where the fence is 30 degrees and use it to ensure your angles. It's a bit slow, but so is kumiko in general.