I doubt that a swamp cooler this small would be of much help. And yes, humidity will be an issue if you don't open the windows.
I took my elderly cat on a few trips (she was great company and I miss her dreadfully). I have a small rechargeable USB fan (OPOLAR brand, there are others) that will run forever on very little energy. In the heat of the summer in California (noon temps over 100), I kept the van parked in dense shade, kept the windows open (but not open enough to allow her to get out) and ran the fan, positioned so that she had lots of choices as to how close she could position herself. I checked on her every hour or so, and she did fine.
Keep in mind this was on private property where I was visiting.
I would STRONGLY advise you to NOT do something like this in a public parking lot or public street. Many states now authorize passers-by to break into a vehicle to rescue animals or children that have been left alone. You can be cited for leaving your dog alone for extended periods, and your insurance company may not pay for the damage caused by the rescuer. Be sure you know the laws regarding this for any state you plan to travel through.