The compressor is a considerable load on your engine. In order to stop the engine from bogging the A/C usually has a way of raising the idle by letting more air past the throttle body. This function is where I think your problem lies.
This can be a separate solenoid specific to the A/C system with a (fairly hefty) vacuum line to both before and after the throttle body. I've seen this solenoid referred to as an A/C Idle Up Vacuum Switching Valve (VSV) in some manuals, but Chevy's terminology may be different. If it is a separate solenoid, there will also be a separate idle-up solenoid for the power steering as well as for large electrical loads (headlamps, large electric fans, rear defroster). You should find this solenoid, verify that it's not working and replace it.
The alternative, I guess, is that there is one ECU controlled valve that performs the idle-up function for all the systems that require it. In that case, unless your engine also bogs at other times, I would suspect the electrical pathway that informs the ECU of the compressor operating.
Please update your question as you find out more, and I'll add to this answer.