Sorry about not being clear. What I am trying to do is run the wire vertically from the attic above, down the inside of an insulated wall to where the outlet will be. This portion of the house is on a slab, and therefore, I have no access from below.
When I mentioned the horizontal framing members, I was referring to the 14.5" pieces that were installed between the studs(I believe they are supposed to serve as some sort of stabilizer or fire prevention method). I know I have encountered them in my home before. Since the other side of this wall is in the garage, I know I can remove some drywall for access to the horizontal framing member, and drill a hole vertically through it to allow my snake, and then sheathed electrical cable to pass through it.
I am imagining a difficult time snaking the wire between the insulation and the drywall for a distance of about 7 feet (from the ceiling to the outlet).
I will cut a hole for the receptacle box, and try to get my snake to it from above the ceiling, through a hole in the top plate. Assuming I can get the snake down to the receptacle hole, I will attach some string/line and pull it back up to the attic. I will then attach my sheathed electrical cable, and pull it down to the receptacle hole in the room (I'll have already pulled the sheathed electrical cable into the attic, hence the reason for sending it down to the receptacle).
I haven't had to snake down an insulated wall before (fortunately, all the post-construction wire runs I have done have been in interior walls), so I was looking for any tips or tricks someone could suggest for preventing the snake from hanging up on the insulation in the wall.