How do I run 6/2 romex through an inaccessible area?

[coburn333]

DIYer here. I'd like to install an electric car wall charger myself. It calls for 6/2 romex which I have. Unfortunately my panel is in my basement on the opposite side of the basement from my garage. AND my basement is finished (drywalled ceiling). The finished walls are metal studs and they do run about 3 inches from my poured concrete foundation because I have a french drain. So I have that 3 inch gap I could easily fish the 6/2 through, but its a 20 foot run where I can't access the wall. See the first pic. I'd really like to figure out how to do this to US/Pennsylvania code and not burn my house down. Options I'm contemplating:

1. Clothseline the 6/2 across the 20 foot span with staples holding the 6/2 on each end. I highly suspect this will not meet code.
2. Support the line in the middle with a rope. See first pic.
3. Figure out a way to snake the 6/2 line through the existing grommets in the metal studs. I'm not sure how to do that with all the insulation in the way over the 20 foot run. See pic 3.
4. Support a 2 or 3 inch white PVC pipe on each end of the 20 foot span and run the 6/2 through that. I'd use the grey electrical conduit, but I think its flexible and would really sag across the 20 feet.
5. Fish the 6/2 through a slot/cavity that I found on top of the metal stud header. See pic 5. This cavity is a 2x6 on one side, dry wall on two other sides and the metal stud on the other side. I believe there are cross-sectional area % codes for when you run romex through conduit. This cross-section area is ~1.8 square inches or 11.5 square cm. My fish went through this fairly easily, but I don't know if its OK to route it through a cavity that has dry wall, wood, and metal stud on the sides.
6. Pic 4 is the original wiring which is boxed in by that 2x6 on the other side, the ceiling joists, my concrete foundation and a header board on the foundation. Its pretty tight in there, but I might be able to get the 6/2 through there as well.