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E=IxR.
If locked, your belt would squeal. Excessive load, same. Your clutch fuse would blow.
Compressors are oiled once, then when you open and drain the system, do it again. No dipstick.
10V where? Engine has to be running to engage A/C, so if 10V at battery, you have a charging system problem.
Do your headlights draw it down? Again, where are you measuring the 10V?
The battery will provide enough power to arc weld. Dirty cables? Still, 14+ off the alternator, not 13. Load like I said would blow the fuse if enough to drop three volts, E-IxR.
The more I think about this, especially since I had a clutch coil go bad this year....the maximum current to the clutch is small. Windings are small. Just enough to pull the plate in on compressor shaft (my winding opened when it got real hot outside...corrected when cold).
I think your alternator has a problem. Or the idle kick for the AC isn't working.
The load when the mechanical AC starts (not electrical) slows the engine down, drops your 13v charge off....and battery is probably shot, too (or not even getting enough charge) to not hold voltage up when the charge effectively goes away when idle drops.