Kraft facers would all but guarantee that a problem gets started. Kraft facers run about 0.4 to 0.5 perms, which is already about as low as you can get away with for allowing the foundation to dry, which means you wouldn't be able to use foam against the concrete, creating wintertime rot conditions at the studs. Unfortunately your inspector is unwilling/unable to do the math on it, but adding a vapor barrier on the interior would add sigficant mold & rot risk from ground moisture to both the studwall and the foundation sill. What makes sense above grade just plain doesn't work for a foundation wall.
Best/cheapest all-foam solution would be to go with 4" of unfaced bead-board EPS (not XPS) for about R16. The fattest you could go with XPS is 2", for R10, and it would cost about the same or even slighly more. Use multiple layers taping the seams as you go, and overlap the seams by 12" or so for best possible air-tightness. If you go any thicker than 4" on the EPS it too will be come to vapor retardent as well.
To save on depth, use 1x furring (or even cut down half-inch ply if you're fighting for minimum depth) through-screwed into the concrete with 5" screws, hang the gypsum on the furring. The point furring & screws rather than relying on glue to hold both the foam & wallboard in place is a fire-code issue- the long screws continue to work at temps that would cause adhesives to fail. But gluing the foam in place as you go makes it easier to assemble.
With 4" foam + 1x (3/4" true thickness) furring, and 1/2" wall board the finish surface is 5-1/4" out from the concrete.
about the same as 2" XPS + 2x3 studwall solution:
With 2" XPS + 2x3 studwall + half-inch furring the finish surface is 5" out from the concrete.
With 2" XPS + 2x4 studwall + half-inch wall board the finish surface is 6" out from the concrete.
Best/cheapest all-foam solution would be to go with 4" of unfaced bead-board EPS (not XPS) for about R16. The fattest you could go with XPS is 2", for R10, and it would cost about the same or even slighly more. Use multiple layers taping the seams as you go, and overlap the seams by 12" or so for best possible air-tightness. If you go any thicker than 4" on the EPS it too will be come to vapor retardent as well.
To save on depth, use 1x furring (or even cut down half-inch ply if you're fighting for minimum depth) through-screwed into the concrete with 5" screws, hang the gypsum on the furring. The point furring & screws rather than relying on glue to hold both the foam & wallboard in place is a fire-code issue- the long screws continue to work at temps that would cause adhesives to fail. But gluing the foam in place as you go makes it easier to assemble.
With 4" foam + 1x (3/4" true thickness) furring, and 1/2" wall board the finish surface is 5-1/4" out from the concrete.
about the same as 2" XPS + 2x3 studwall solution:
With 2" XPS + 2x3 studwall + half-inch furring the finish surface is 5" out from the concrete.
With 2" XPS + 2x4 studwall + half-inch wall board the finish surface is 6" out from the concrete.