Anthing above-grade section you'd be OK to use either faced or unfaced batts if you're just fattening out the 2x4 studwall with another studwall. Below grade you'd need to use unfaced batts, and putting 1" of unfaced rigid foam (ether EPS or XPS) between the studs and the concrete as a capillary break and to keep the temp at the stud edge above the dew point of wintertime interior air.
Kraft facers are not true vapor barriers but are semi-impermeable vapor retarders, running ~0.4 US perms. That's sufficiently vapor-open for seasonal drying, yet vapor tight enough to limit wintertime moisture accumulation in moderately cool (but not true cold) climates. While stacking a pair them in your plan would be OK, it's still less than ideal. With just the single vapor retarder placed at the mid-point of the total R it will have about ZERO condensing hours every winter at the facer in a Shillington PA climate, and thus won't become a condensing surface.
To get better thermal performance out of the double studwall, stagger the studs of the new with those of the old as a thermal break. If there's space, inserting even 1/2" of rigid XPS sheathing (or even foil-faced iso, if above grade) between the new & old studwalls and detialing it as an air barrier (mastic or tape sealed seams, caulked or foamed edges, etc.) is even better. Far more moisture problems are a result of air leakage than from vapor permeation. Caulk the gypsum to the new studwall as you go, and air-seal every electrical or plumbing penetration too. A square inch of air leak moves as much wintertime moisture as a whole wall's worth of vapor permeation through latex-painted gypsum.
