Would it be possible to half lap the boards? Since there is no structural need to it, you could cut away perhaps 2/3's of the material thickness for each lap. This would allow for variance in the rough cut boards. Even something as simple as cutting a bevel on the boards would eliminate being able to see through or behind the boards, so you wouldn't need any paint or colored materials behind the boards and open up other avenues for backing materials. Rounding over the bevel on the surface can work to hide small gaps if they develop over time. If the material is very green it could be helpful to let it dry out very well before hanging it, as you could end up with splitting, warping, and other problems beyond shrinkage.
If the paper-backed insulation is tacked down well, you might be able to seal up the edges instead of covering it with entire sheets. Some sort of tape, glue, resin or similar to seal the edges could be sufficient. You could also possibly use some sort of cord or other packing material if you go the half lap method, similar to old ship building methods. You don't need a
water tight seal, just
enough to minimize and filter any potential airflow.