There is nut that is sometimes there/sometimes not on the center shaft for the clutch. I've got a Harrison R4 guide that shows NO bolt in the assembly/disassembly instructions; but the M-B guides DO show a bolt there. You can rent/borrow the clutch tools to set clutch plate depth and adjust the R4 clutch from Autozone and/or O'Reilly's. I recall one or two forum members who have reported losing that center bolt and then having clutch failures.

But I've also read that noise is a common complaint with R4's, so what you're hearing might be normal-ish.

New R4's can run about $300, so it'd be a pricey "preventive maintenance" swap if it's currently cooling okay. If you go that route, a complete flush and re-fill with the proper quantity of oil would be prudent since lack of lube due to previous oil loss may be the root cause for the noise. If your daughter's driving it and it's out of your control, a few hundred dollars may be small price for piece of mind, though.

Are there any traces of oil leaks on the A/C system? Maybe drop the passenger panel to look at the TXV near the evaporator?

Some people might say 134 in a W123 that "works well" in California is an abnormal condition. Have you put gauges on the system to check pressures?

Are you sure the compressor is making the noise and it's not from one of the other pulleys due to the extra load when the compressor is engaged?