Since the two systems AC and trans are isolated and unrelated the
only common between is the serpentine belt. When the AC clutch
kicks in it is causing more a demand on the engine's torque, than
normal. So the engine is being taxed hard. This is a complex
problem and I am telling you now I do not have the answer, but I
have a few suggestions which may be a solution. 1) the AC
compressor could be working harder because it is low on r134a
refrigerant. 2) your trans fluid is low. Check it with the engine hot,
and if it needs more fluid add Lucas Transfix. If it does not, do NOT
or fill it. That will cause leaks 3) check the tension on your
serpentine belt. There should be an indicator on the tensioner to
say if it is within working limits. BUT if you see the belt shaking and
not moving smoothly then the belt does need to be replaced.