I had this problem exactly on my 95 855.
When the ac became faulty, the compressor clutch would not engage unless engine is switched off to let the coil cool. To avoid working the compressor clutch too hard, the controls could be set to minimum coolness and it would work all day.... OK maybe, but not very cold.

Try this....
With the engine NOT running, disconnect then wire to the compressor just before it enters the comp body. Now use a thin jump wire (with an inline fuse), connect one end to the compressor and the other end to the positive side of the battery thereby engaging the clutch. It should click on like it does when cycling. Leave it like this for 20 minutes or so. Then disconnect it then connect it again and if it doesn't engage again, this is the fault, you've got a weak or overheating clutch coil. Other possibilities are a poor earth on the compressor or not enough voltage at the supply wire leading into the compressor (the one we disconnected for the test).
Don't forget that for this test we used 12v to engage the clutch and when the engine is running, it is capable of supplying 13.8v or more.

I had this problem and made it work again by earthing the compressor body with a battery strap bolted to a spare lug hole and the other side to the body of the car (not to the engine). Secondly the feed wire to the compressor now activates a relay (the type for wiring foglamps) and the relay supplies current to the compressor straight from the battery terminal (over 13.8v). Make sure all the connections are securely crimped and use a decent guage wire for the supply to the compressor clutch. And make sure it is fused...5A should do it. I measured it at less than 3A.

The proper fix i suppose is to change the compressor coil or clutch and at 120k you might as well change the compressor at the same time but this must be expensive.

But before you do all that, theres a overheat sensor on the back of the compressor. You can see the wire leads into the compressor body and then out again before going into the clutch at the front. You can simply bypass this by unplugging the connectors and er....bypassing it. Maybe its this causing it to cut out. Previous posts have suggested it is safe to leave out.

Be extremely careful when working near the drivebelt when the engine is running. There are no covers over the pulleys and you can instantly have your hair ripped out or lose a finger.