I wouldn’t recommend it. . You might make it, but the wear on tear on a small car like that will likely result in a lot of problems in the future. Not worth the risk. Instead, price out Budget, Penske etc ‘Rent a Truck’. A 12 footer would easily handle a one bedroom apartment. You might could even fit everything into a pick-up-truck. The price to rent trucks in these modern, computerized days varies a lot from week to week. If you happen to live in a place that has a lot of trucks parked and waiting to be rented at the current time, the rental rate can be surprisingly low. If you have two adults you can convoy, one can drive the smaller car, the other the truck. Gamble and enjoy the buffets at the casinos along the way, make it a short holiday.
The other problem you’ll have to deal with is that when you import a car from another state into California, getting it registered with California plates can be somewhat expensive. They do that to dissuade folks who live in California from buying a car in another state to save some money. Suggest you phone up the Calif DMV or at least search their website to see what you’d be looking at. I’d guess it probably makes more sense $$-wise to defer buying a car until you get to California. No problem, you’ll be driving the rental truck. I’ve heard on the radio the California DMV is swamped right now with people trying to get some sort of ID they need to fly on an airplane, so expect some difficulty if you try to phone them.