Ok, gotta ask, since you lobbed over a very vague question, albeit directed at the benefit of being white (hope that doesn't come off as racist, lol), and not necessarily the process it would take to get it white.
What is the camper, what color is it now, what is the siding made of and what type of paint/how do you plan on "re-painting" it?
Asking because, 99% of campers, especially older than say the last 5-10 years ARE white in color. (obviously yours isn't)
Most campers that aren't white, aren't a very dark color. Typical shades are beige or grey.
Most darker colors on campers are not paint but rather decals.
Lets talk paint. You talkin a paint brush, roller and some Rust oleum? Or a good single stage or 2 stage urethane auto paint? And what is your level of experience painting vehicles? This is the main reason that your thought sounds like the effort will not be worth the gain.
I'm middle aged and with experience in autobody and painting and quick thoughts, IF I had the time to work full time on something like this, would be "about" 2 weeks worth of full time work. And probably $1000 min in paint materials assuming you have all the tools, gun, big compressor AND an indoor place to paint it, or at least out of the sun. (outdoor paint jobs in the summer are another whole discussion)
Back to materials, that $1000 is bargain priced paints. Double that if going for long lasting quality (you are in fact planning on exposing the camper to the worst possible conditions for paint for extended periods of time....or why would you go through all this effort for 1 or 2 months trip)
In short, need to hear your thoughts on what and how you would accomplish this, to decide whether it's simply not feasible or super ridiculous to attempt.
And to the original question, I've lived in AZ in the summer, and not sure if the color lowered the temp from say 140deg to 125 deg, it would make a discernable difference. Being in the shade would, as would, not trying to air condition a canned ham in a convection oven.
