Proper way to route new wire through existing exterior wall
If I were you, I'd invest in a fiberglass rod or wall-fish tape. The process to get the wire down is that you will cut a hole for your drywall box and then use the rod/tape to push the wire down inside the wall (use electrical tape to lash the wire to the tape/rod). The hole should provide you enough room to get your hand inside and grab the wire. Then disconnect the wire, pull your tape/rod out, put the wire into your box, attach the box to the drywall and you're set.

The kraft paper on your insulation should be on the exterior wall so it shouldn't be an issue on an inside wall. The wire will be inside the insulation and loose.