California Hwy 1

NO!!! That portion of 101 has many hairpin turn, there are sections where there really is no discernible shoulder. Going South you will be on the cliff dropoff ocean side. Oncoming traffic will often move fast, and service trucks do take that road making deliveries to Big Sur. We drove our 38fter with toad North a couple of years ago, even though I knew better...but I forgot just how bad the drive was. Going North, with the mountain cliffs on your right, and roots and rocks protruding onto the road access I made it holding my breath for about 12 miles. It is a beautiful drive, that the driver cannot enjoy. I would suggest taking 101, a nice lovely easy drive, from Monteray Bay to Morro Bay, stopping at some of the many wineries or vegetable stands. Base camp in the Morro Bay area for a day or two and take the car to Big Sur.
Happy Trails.