A hot day with AC ON wont necessarily lead to engine heating
IF the FAN is not working its a problem, but if FAN is working its not a big issue, but do take care of coolant and engine oil levels as a caution every 2000 kms or so just to make sure the issue is not related to engine
Another possible issue is a resistor or a POT that might have gone wrong, if that goes wrong the FAN speed wont adjust itself based on the current input to the FAN, so whenever there is a signal to switch ON fan at a specified speed, the lack of resistor POT to control the FAN might lead to a short and excess current might pass through the circuit causing FAN to work in a low voltage high current situation, this is not ideal and can cause unwanted sound from FAN and possibly vibration as well as the compressor too are connected to this circuit leading to low current or voltage in compressor leading to vibrations,
SO my suggestions are to check the thermostat and the Resistor POT that controls FAN speed, a thermostat error usually should be detected by OBD but Resistor error might not show up on this,