I have to agree with both the previous posts. This is way too much to see and drive, in the short amount of time you have allowed. Having just returned from Yosemite last week and the Grand Canyon last month, my suggestion would be to drop the Grand Canyon from your list. It is "Grand" in every sense of the word, however, its just a big hole in the ground! It is massive and huge and awe-inspiring, but you don't get any real idea of just how big it is, because of its enormity and the fact that you are looking down into it. That fact alone, that you are looking DOWN into it, as a human, makes everything seem somewhat lesser than it really is. Though that's just my humble opinion.
On the other hand, being at the bottom of Yosemite Valley and looking up at those massive granite cliffs, gives the true sense of wonder and awe that you feel so small in this vast world.
As I said, just my opinion.
If you are still doing the Grand Canyon, take I-15 North to I-40 East almost all the way to Flagstaff then head north to the park. Come back the same way but go north on Highway 93 in Kingman to get to Las Vegas. See below for where to go after leaving Vegas.
If you cancel the Grand Canyon, Las Vegas would still be doable, its a long HOT trip in the desert one way and back. You could always use one of the days you save skipping the Grand Canyon to stay one more in Vegas, if you want to. Hoover Dam is pretty neat if you make the time, its about 30-45 minutes from Vegas. Go early, as traffic backs up quickly at the security check point. Be warned the officers, check the ENTIRE motorhome, top to bottom front to back, inside and out, to make sure no one tries to blow up the dam.
If you're still bent on going to Vegas, take I-15 from LA to there, then on the way back down I-15, take Highway 58 at Barstow, CA towards Bakersfield. If you want quicker, more boring drive take Highway 99 from Bakersfield to Fresno and Highway 41 into Yosemite. Another scenic route would be Highway 58 to US-395, up the eastern side of the Sierras to Highway 120 and into Yosemite. This does require a trip over Tioga Pass up to almost 10,000 ft (3,048 m) don't know if anyone in your party (or the motorhome) can handle that, though.
From Yosemite, take highway 120 West to I-205 to I-580 straight into the Bay Area. After San Francisco, if you have time, take Highway 101 all the way back down to Los Angeles. Much nicer drive than I-5 and only an hour or two longer if you don't stop on the way.
Places to stay? Hopefully, you've already made reservations in either Grand Canyon or Yosemite. They book up as soon as they become available inside the parks. Grand Canyon isn't as bad, there are RV parks just outside the park. Any RV parks outside Yosemite can take up to an hour to get into the valley.
In the Bay Area, there are very few RV Parks near San Francisco, you will be driving at least 30 minutes to get there. Plus with an RV the size of yours be VERY VERY careful driving around the city. Many small, narrow streets, and some extremely STEEP hills. Best advice don't drive THROUGH the city, stay mainly on the Embarcadero near the waterfront.
Good Luck, hope you enjoy your trip. It may be brief, but it will be fun!
Regards,
Tom