Why not crate them for the trip?

Crate them and keep them in your car with you. I believe that in most states it is illegal to carry any living creature in a towed vehicle. At least that is the law when travel trailers. If kitty is stressed pick up some rescue remedy, that will help sooth her nerves and calm her a bit.

I am distressed that you are considering "rehoming" a beloved pet. Why on earth? Is she not one of your family?

Anyway, I would crate them both and use rescue remedy. Your voice and presence will comfort them so much more than anything else. Keep them nearby.

Crate her and get a harness for the dog. The cat will do fine in the back seat in a large crate and you can experiment as to whether she likes a view or not. When I recently traveled with my Mom's cat we discovered she liked being elevated so she could see out the windows. When placed below the windows she wouldn't stop meowing. Also carry a towel or small blanket so you can cover part of her carrier and keep the sun off her.

Moving is not an excuse to rehome your pet so I hope there are other reasons behind you considering do so.

On our last move my DH drove the van and I followed in the car. He had one dog and I had the second dog, along with 8 birds! Consider a second vehicle if you're trying to fit everyone in the van.
Forget putting them in a towed car since your pets won't survive the heat and wind blowing thru open windows will add to their stress.

Start doing short trips now with Claudette in a large kennel in the car. You want it large enough that it can hold a small litter pan and some water. If she likes high spots, consider creating a "kitty hammock" out of a large dishtowel and some snap rings so she can elevate herself off of the kennel floor. Put her in the kennel and give her a treat. Drive around the block and give her another treat. Do that every day for a couple of days, and then lengthen the trips. Go through the drive through at a fast food joint and have her hear the speaker and smell the open window scents. (This is the only kind of food you can grab while on the road with pets.)

While you are training her that travel doesn't always mean going to the vet, it's also a good time to try to train her to become accustomed to a harness and leash.(You should also make sure that both pets have good current ID listing your cell number and you old vet's number, just in case.) That's a safety measure in case you stop at a rest stop and need to open the door to refill her water or something. If she's wearing the harness and tethered with the leash, even if she darts out the kennel door like a banshee, she can't go far. This isn't a bad idea for your dog either. Startled dogs can slip away from you quickly. We had a rescue group in transit stop at a local rest stop and two of the dogs slipped away from the group on the potty break. Despite a massive manhunt, the dogs haven't been located yet. Don't take any chances that this could happen to you.

Map out your travel route, with your stops at pet friendly hotels planned for not more than 8-9 hours of driving. Keep the harness and leash on both animals, or place them in the bathroom with the door closed while you are going in and out of the door. The harness and lease doesn't come off until you retired 100% for the night. You could also ask your vet for a light sedative if she remains a poor traveler. It's not ideal to keep an animal under sedation for long repeat periods, so if you chose that route, you'd want to plan to push on for as many driving hours as you could safely do so.

PS, you could also consider having a family member in Denver fly the animals to you after you've made the trek with your possessions in the car. The weather should be cooled off in September and the airlines be allowing pet travel again. Some airlines are better at this than others, with Delta probably being one of the better choices since they are based out of Atlanta. My SIL's cat Stinky Boy flew from DC to LA when she moved there, and then from LA to Memphis when she moved in with us for a while, and then from Memphis to DC when she moved back. And then from DC to LA back to DC again when she couldn't decide which boyfriend she wanted to be with. He practically had his own frequent flyer account! She used Delta, and was very complimentary about their process and pleased with the cost vs. car travel hassle. You will need a health certificate showing that the animals are up to date on their shots and healthy enough for air travel and the airline requires an approved crate with water and food available. I think it's about $300, but if you have a poor traveler despite all of your training trips, then it's worth it to minimize your and her stress.

What's "rescue remedy"? When my parents moved & I was a teen, we got pills from the vet that were supposed to make our pets go to sleep for a few hrs. I don't know what it was called, but it worked for the dog and 2 cats. The 3rd cat fought it all the way. Also he did not like to be in a crate - it made him more panicked. I finally had to let him out and hold him. I may have to move again with 3 cats and a dog. Two cats hate car rides and will not tolerate crating, so I am interested in the advice you all are giving the previous poster.