1860 Timber Frame Exterior Bathroom Wall

[TechEditor]

House was built in 1860. Re-doing the Bathroom. The exterior wall was original Horsehair Plaster, under Masonite wall board and drywall. Doing a complete gut and re-do. When I got to the lath and removed a few strips, I only had 1-1/2" of room for insulation. The lath was nailed to strips of wood, 1-1/2" thick and very wide. What I want to do is remove the lath, and the nailing strips and insulate. What kind of insulation? Rigid? Roxul? Spray foam? I drilled a hole in the wall just to see what weas under the wide boards. The wide boards are 1-1/2" thick and under them is what appears to be black tar paper. So, I can assume I am at the outer wall boards, and this wall is solid with no cavity to insulate except for the initial 1-1/2".
I am open to ideas here.
See attached images.

Greg

[mrcaptainbob]

Maybe add furring strips to the 'studs' to build it to a 3" depth? That would allow for standard insulation, but would detract from floor space. May not be an issue for just the one wall.

[HayZee518]

what's the thickness of the studs?

[TechEditor]

The plank "studs" are 1-1/2" thick