I would check the compressor itself - is it locked-up. It shouldn't matter though, that's what the relay is there for to turn off the compressor if it's locked-up.

Try unplugging the compressor and see what happens, if the clutch is drawing too much amperage? Although this scenario should blow a fuse or the relay.

The only other thing that comes to mind would be pretty far fetched, POSSIBLY... the RPM signal being grounded/shorted when the compressor is engaged? This would definitely shut the engine down. But like I said that would be very strange indeed.

Try what I've suggested and let us know what you find.