Let me address your questions.
The PCM is the Powertrain Control Module. The BCM is the Body Control Module.
For the most part, these modules do not fail when it comes to AC problems. Although they can fail. The Modules communicate between each other via serial data.
The BCM handles the Blower Fan Circuits and the Pressure switch and then tells the Powertrain Control Module via serial data to adjust the engine for the load the compressor is going to put on the engine.
They simply look at Your AC High/Low Pressure switch and if they see the AC pressures within a safe range, will allow the compressor engagement.
So the Body Control Module tells the PCM....OK we have a request for AC, and the pressures are OK acording to the Pressure Switch input to the BCM. THe PCM then Grounds the primary side of the AC Relay, and this energizes the Compressor.
If the pressure is low in the AC system, The PCM will not ground the relay.
A shortcut would be to bypass the pressure switch and trick the modules into thinking the pressures are good. Then attempt to turn on the AC at the Interior switch.
If the Relay energizes and the Compressor Engages, then you either have a bad pressure switch, or the system is low.
Since you stated that you already bypassed the relay and the compressor engaged, then I'll assume that the compressor clutch is functioning.
A set of AC gages from Autozone should tell you if you have pressure in the system.
Keep in mind, that if the system is empty, there is a chance you have a leak somewhere, and you will need to repair that leak and then evacuate/vacuum the system to remove any moisture before recharging the system.
So by all means, temporarily jump the light green to the dark green wire at the pressure switch just to see if you can trick the PCM AND BCM into energizing the compressor clutch relay. If everything seems to work with the switch bypassed, remember that this pressure switch is there for a reason. If the compressor runs under conditions of very low or very high pressure, the compressor can fail.
I hope this information helps.