My guy is an excellent traveler. When on long trips he has a harness and leash that's hooked up to the rear seat belt. It gives him some freedom of movement. His covered litter box goes on the floor behind my seat and his crate goes on the passenger side back seat with his bed on top where he sits and watches out the window. On the floor behind the passenger seat is water and dry food although he only eats after we've stopped. Hotels/ motels that accept cats are very easy to find. I've never had a problem with that. But doing that for a month would be very hard on your cat. Take her favorite toys and make playtime at the motel/ hotel a priority. I'd suggest taking an overnight trip several hours away for a weekend to see if your cat would even be comfortable with a trip.

Considering that you're going to be going to museums and other places and leaving the cat in the motel/ hotel or car alone I'd seriously think of leaving him with mom and dad.