As others have also stated...we would not! Doing so would spoil the trip. The longest day driving we've ever pushed was 10 hours with both sharing driving duties. But, no matter what time of year--we always park before dusk. You have to consider you'll miss seeing the areas you travel through in the dark...for us that is a deal breaker! There is a lot to see between your area and the west coast!
I would not consider it safe, especially if driving the rig is new to both drivers. A Class C isn't much larger than a big dualy with a long bed but...it is different, more weight, more affected by wind, etc. It will handle differently! Extra attention to details will be more tiring.
If you only have a week to take, going from coast to coast and back it's just not too doable that far (my opinion). I would hold that trip for sometime when you can take a month off and go somewhere much closer this trip.
You really won't know what you can handle until you try it.