To prevent over heating turn off your Air conditioner for the next 10 miles !

If all things were equal it should stress the engine slightly more than a conventional car because of the conversion loses, but since I believe the Gen III Prius is a variable speed compressor maybe not. I believe the Gen III Prius also has a variable speed water pump, not sure about that though, but it is electric, so maybe it is.
OLD cars benefited the most with the AC off, many newer cars have electric radiator fans and some of them have two fans, sometimes only one fan will run to cool the engine, but if the AC is on both fans will run. My Miata is like that. So in the Miata, running the AC may help in cooling the engine because your running two fans instead of one. YMMV of course.
NO car should overheat ever, if the cooling system is in good shape.
If you think you may overheat, one of the best things to do is to turn on the heat as high as it will go. Obviously the heat comes from the engine cooling system and the interior heater just became another cooling radiator for the engine.
And no, I have never seen a sign like you have in California.

 

I think the sign should say
"If your vehicle has less than 140hp, please turn off the A/C to prevent overheating."
In my old 1991 Acura Integra (130-140hp), turning off A/C is like turning on turbo!!!

 

The signs are there because when climbing long grades you are dumping a lot of heat into the cooling system. That heat has to be extracted through the radiator. When it is hot the radiator has a much harder time getting rid of the heat. With the A/C running you complicate this by making the air going through the radiator hotter because the condenser coil (the radiator to get rid of the heat from the A/C unit) is mounted in front of the radiator and it heats the air more. The only was to tell if you need to turn off the A/C is by monitoring the radiators temp (no temp guage on the gen 3 Prius). If you had a scanguage you could monitor it.

 

Also, sometimes the traffic is very slow up these long grades, so the cooling is not as efficient as it is at higher speeds.

Usually, well-maintained vehicles have few problems.

However, one ccasionally sees overheated vehicles pulled over to the side of the highway. Usually they are older vehicles, perhaps with leaking pressure caps, and maybe using just plain water for coolant.

 

My informal polling of overheated cars on the grapevine on hot days are usually GM, Chrysler, and Ford. I've never seen a Japanese car with the hood up.

 

I'm contrarian on this. For all Prius years, if your engine cooling loop is functioning properly it is comfortably oversized. However the HV battery can be doing big energy transfers on inclines and its air-cooling system is not necessarily oversized. That cooling system is improved when your A/C is running.

So when the road sign says turn it off, I turn it on.

 

That's logical