For the first cut I used a shorter, metal-cutting blade in the jigsaw. (The larger teeth of a wood blade seem to get bound up in the fibers of the cotton insulation. The finer teeth of the metal-cutting blade had less of a problem.)
The 1/4" birch scrap promptly fell into the gap behind the paneling :-(. We cut away the cotton and foam insulation with a utility knife, leaving the 3/4" exterior plywood exposed.
Ouch, the foot of the jigsaw mars the wall a little. I think it can be repaired with some light sanding and touched-up with shellac.