Many people drive this route, it is all main roads and there are several routes you can take. Many of the roads will be motorways but not all. And main roads in mountains will not be straight.
If your driver has experience with driving in the weather you happen to get, in the kind of landscape you will be in and your car or van is fitted out for the weather, you should not have problems. Chains for when you get stuck in snow at least.
The biggest if is your driver. Snow and icy roads are not guarantied but when they are there is need for a driver that has experienced these conditions before in a vehicle that is like the one you will be driving. Having a driving license from Washington state does not tell anything about the driving experience of the owner.
If your driver is experienced in winter driving in mountains, it might be worth it. If not, please consider taking the train instead, or a coach of course.
The only real reason to travel in your own vehicle is wanting to make stops on the way. If your only stops are in towns or cities with railways stations or stops in the bus service you prefer, you better of taking that kind of travel. Not because car travel is dangerous, but because not having to drive is likely to make the travel much more enjoyable for the driver.
Google maps predicts a 4 to 5 hour journey by car, but if you do hit winter weather that may well become much longer.
Train travel will take longer, roughly 6 to 7 hours. With one change of trains.
On the positive side, those changes of trains are all in cities you may want to visit, so you could break your journey there, without having to find parking.
Bus travel will take longer than car travel as well, but is likely to be cheaper than renting and all other costs.
In a comment is mentioned that renting a van can work out cheaper. Do compare with the actual train or coach prices and remember that fuel is/can be expensive compared to what you are used to. Public transport often has cheaper rates for young people, car rentals are often more expensive if young and/or inexperienced. Or not available at all if you only have had a license for a short time.
And if you are traveling one way, consider the drop off fee on the van, which can be substantial as you will be in a different country.
On top of that may come the price of motorway tolls, done in a once a year sticker payment which may already have been paid for the car, but if not, you will have to pay it.