With the LTZ you get ACC - automatic climate control. It will run the AC compressor without turning on the AC light. You might be experiencing the extra draw on power when it kicks in. To test this theory, next time it shudders turn your fan knob to the "OFF" position to shut down ACC completely. If the shudder stops then it's the AC compressor. If not then that's one component/system you can eliminate as a suspect.