I'll second the idea of using a small hard-sided cat carrier. You'll want something that can protect Athena in the event of a collision. Even hard stops can send a small animal flying around; a small carrier limits the amount of space she'll travel. If you don't already have one, look for one that has a little tab for a seatbelt to run through.
Take a look at using some sun shades to keep the sun from shining directly onto the carrier. You can get little mesh screens that suction-cup to your windows and you can still see through them (not perfectly, of course). Placing the carrier in the middle of the back seat will be the best protection from collisions and the sun. You'll also be able to glance in (when you're at a stop, of course) from time to time.
Definitely keep the carrier off the floor. Initially, the floor seems like it might be the best place: heat rises, carrier can be wedged in nice and snug, it's down low so the sun won't be as much of an issue... all that. But it doesn't work. The amount of heat coming off the road can and will kill.
I use thermometers with probes that I can put inside the carrier with my hedgies to ensure they're at a good temperature. I don't know if a chin would try and chew the probe or the coated wire... I'll leave that to someone with greater chin knowledge than my own. Ensure you're A/C is working well before you begin.
I'd keep chin-safe cleaning supplies and plastic bags near the carrier to mop up any bathroom messies. With luck, she'll sleep, so it won't be a concern. But you don't want to find yourself with a pee-filled carrier and only your t-shirt to clean it up.
When you pack your car, do your best to keep her supplies on top. I'm assuming, with that distance, you'll be stopping and setting up things for the night at least once. You'll want to get her all settled first, then worry about where you put that pair of shorts and your toothbrush later... especially if she's had any difficulties along the way.
Given two international border crossings, make sure you've checked with authorities at both entry points to ensure you have whatever paperwork needed on hand. It could be that they require a letter from your vet within the last 30 days... you'll want to get that settled before setting out. I remember with the hedgie train into Canada, they were okay with a Niagara Falls crossing, but not so good with Detroit/Winsor nor Port Huron/Sarnia. They had a vet on duty at NF; not the other places. Hopefully, they aren't as picky when were talking about one chinchilla instead of ten hedgehogs! But check it out in advance, eh?
