Definitely DO NOT apply battery voltage to the low/high pressure switch.
The compressor is controlled by the Base module. This is a large power distribution relay that handles the fuel pump, oxygen sensor heater, and maybe a few other power users. The signal for the base module to turn on the compressor comes from the pushbutton controller THROUGH the low/high presssure switch (on drier). It is a GROUND signal. If you apply battery voltage you will find the weakest link in the pushbutton controller. It burns a trace on the PC (printed circuit) board. I have seen it many times.
What you need to do is check the voltage at the switch. It should be low 0.1v when A/C is on and it will be high (12+v) with no signal from PBC.
You will need to verify these voltages at the low/high pressure switch for the above activity and for my accuracy. There are a few models that do apply 12v to the a/c comp module. I am pretty sure its a ground in this case, but the wiring diagram doesn't tell and I haven't written it in (as I usually do).
Verify this circuit by identifying where each of the two wires go. One wire will receive a signal that should change if you turn the A/C off and on (of course it may not be working). The other wire should come from the Base module and I think that it should have batt voltage on it. It will disappear it you unplug the base module. The base module is in the CAN box along with the engine management controller (LH) and the ABS controller and the Cruise/idle controller).
It will be very hard to do any futher testing without some special tools but both controllers should have self diagnostics. This can be evaluated without special tools if one has one of these devices: logic probe, LED, or DSO (scope). It you feel up to it let me know and I'll provide a description for testing. Improper use (or misunderstanding) of those instructions could be very costly so I won't print them here.
Owner 24 bay BSC
Bosch Master, ASE master L1
26 years MB technician