To clarify, i am talking about foam as structural component, as in a foamie.
Expanded cork board seems interesting but again heavy. A small 40 x 20 inch board 25/32 thick weighs 5.7 pounds

Not an option then
I've looked into wood-based insulation panels,
https://www.pavatex.com/en/products/wall/pavatherm/Seems usable, but about as heavy as the expanded cork boards.
Might both benefit from the mix to control water ingress and pest control, but presume they would absorb quite large amounts
In a more classic stick/ply build i would consider wood wool insulation.
On the google i found something about cellulose foam board that seem to be in the works.
https://www.ecohome.net/news/1465/cellulose-foam-insulation-boardsRockwool is not a natural material though
The base materials are natural, but after the process of turning chalk and rock at 3000 degrees Fahrenheit into rockwool, a hybrid is created that remains in that form forever. It is not biodegradable and will sit in landfills indefinetely.