Wow... That could explain a lot! Time for me to go find a clutch plate remover/reinstaller tool.

I have a clutch remover tool "rented" and am attempting this procedure now (tire and splash guard removed), but when attempting to remove the center nut on the clutch... the clutch is just spinning. I tried to turn the engine on and "engage" the clutch to hold it still, but as soon as I turn the engine off it apparently disengages again. Do I need to prop/jam the clutch plate somehow to keep it from turning while I loosen the center nut? I don't want to damage anything.

Yes, you need to find a way to secure the clutch assembly.

Any suggestion(*) on how to do that would be much appreciated. I've got decent tools, but no pliers/grippers big enough to grasp the outside of the plate. I can't figure out anything in my tool collection that will work. Closest I got was wedging a board in the engine compartment so the clutch plate "struck" the board edge as I turned the center nut. But then the metal clutch plate cut right down into the wooden board (as I turned the wrench) and left a nice plate-shaped divot.

I'm going to need something solid here, since this clutch rotation will need to be secured throughout this entire process.

this and a 20% coupon will make it about $16.00
at harbor freight
12v Impact Wrench
zap it off dude
don't use to put back on
clutch engaged with belt on should hold it for that

I don't have much clearance and I don't think I can get that impact wrench in there (compressor is still on the car). I could probably use a couple socket joints/extenders but they probably aren't rated for impact speed. Still worth a shot, although closest Harbor Freight to me is 45 minute drive.

Also how do I engage clutch with engine turned off? I tried engine on, turning A/C on (to engage clutch), then engine immediately off... but clutch apparently disengaged with I shut engine off.

unplug compressor get a pack of jumper wires with alligator clips {insulated ones}
clip one on 1 spade terminal then on eng some were to ground it
clip other on other spade terminal
to battery +
instant clutch engagement
use a pick or a small screw driver to gently pull up the connector on each side of the
compressor so you don't brake it off 1 on each side
Push down on the belt to keep tension on the belt {if necessary}
small piece of card board between the alligator clips on terminals will
keep any chance of a spark from happing
nut should come right off its only on at 12lbs torque

I would take the belt off after you get the nut off and spin the clutch disengaged by hand to listen for a bad bearing
every time I ever saw metal shavings on a AC compressor
was from the bearing coming apart making the hub rub
and grab it and with both hands to check for side movement

Sorry to say, but belt tension isn't going to keep the compressor from turning. You need a sure grip way of holding it. Or, a really good impact gun that can rip that nut off with a quick braap.

The way I understood his suggestion was to electronically engage the compressor clutch (with jumper wires), and press/tighten belt as a secondary method to put pressure on the pulley of the already-engaged clutch. I'll try it anyway, probably tomorrow morning when I get a chance.