I can't answer all of your questions, I traveled cross country twice in 2014, both times were in cooler weather though. Going from MD to AZ we left in February and the gliders traveled in a RV, in their set up vacation enclosure (secured). We left them inside the RV for bathroom breaks and they were brought into hotels overnight. We didn't disclose them to hotel.
I returned in November of same year, traveling back a slightly different route. This time I was alone with my pets and we all traveled in my suv. I kept their vacation enclosure set up, but they traveled belted up front in a carrier cage, my dog had most of the back seat. The same, for bathroom break, gassing up vehicle, and grabbing a Belvita for breakfast, all pets were left in the vehicle, I had no other choice. I walked my dog for potty breaks, the gliders remained asleep in their carrier. Again, everyone went to the hotel for night, I didn't disclose the gliders to the hotel, but I did disclose my dog all but one night.
I don't believe I traveled through any illegal states, but I would think if just passing through and staying overnight, it shouldn't be a problem.
As for it being too hot to leave in car for potty breaks, I suggest for daytime traveling having them in a bonding pouch inside a small cage/carrier secured, or a small carrier you can take with you.
Pic of small cage carrier setup, you can swap a bonding pouch with a cage pouch that can be zipped up and secured for taking inside for bathroom breaks.
This is how I set up for night stays in hotel, placing the cage carrier inside their vacation enclosure and tying fleece on the top door to prevent accidental shut-in/out. Gliders could come and go, food and water set up, as well as sleeping pouch were kept hooked on travel cage carrier (which would be secured by seat belt for travel time). They had access to their travel wheel, some extra space and toys for overnight.
Other than those options, you could also purchase small, canvas type carrier, or use a bonding pouch, both would have to be secured to seat backs whether used upfront or back seat. The bonding pouch can hang over head rest and use shoulder strap of belt to hold the hanging strap of bonding pouch securely, or use complete set belt to hold canvas carrier.