A car could *potentially* run slightly cooler with the A/C on ONLY because the way our fan works is if the temperature is hot enough to engage the low speed fan, and you turn the A/C on, the fan automatically switches to high speed. But that all changes if you change the temperatures that engage the low/high speed fan.
But the A/C itself will no doubt cause the engine to run hotter. The principle of air conditioning isn't cooling, it's removing heat, and the evaporator absorbs heat from the cabin, pumps from there to the compressor, which heats the freon further because of the heat caused by the friction of pumping and the compressor rotating. That all makes the freon boil which actually makes it release heat more effectively than if it were lower temperature. The condensor has to dissipate a LOT of heat.
The condensor is also an obstruction to the radiator. The thicker the radiator, condensor, etc., the more of a pressure drop you'll see once the car gets moving at highway speeds or so, the electric fan with it's set RPM just doesn't rotate fast enough to overcome it, whereas a mechanical/clutch fan does. With my older 65 mustang, I put an electric fan on and it did great around town, but the faster you went, it got pretty hot because the copper brass radiator was so thick. Took the condensor off (a/c wasn't working anyway) and it did a lot better on the highway. a little off the topic of the specific car but something to keep in mind with cooling systems in general, it could be applicable to ours if we put thicker radiators (like 2 cores) in addition to air/air intercoolers and such.