if the cars getting hot and u missed the hot/cold needle somehow, the A/c trips.
diagnostic charges are around 300-600 depending wat the shop charges.
btw are u saying that the check light is coming on when the car gets over heated?
the check light only comes on if some sensor is maulfunctioning or theres some kinda short circuit.
it has nothing to do with the fuel pump or the car getting hot.
and ya the fuel pump presure can be checked at all times if its weak or not. its not like that , k u can not determine anything unless it dies or its working.