Okay, what you know. The compressor will operate. The A/C relay is good. Bypassing the relay will cause the compressor to operate. If the A/C was working good before you 'cleaned up' the wiring, maybe recheck that wiring.
You are missing either the power supply for the relay, which is the 15A IGN fuse, or the ground for the relay which is in the PCM.
The way the system works is that the HVAC Conrol Module sends a voltage signal to the BCM requesting compressor operation. The BCM looks at the evaporator temperature and if it is above freezing sends a Class 2 Serial Data message to the PCM to turn on the compressor. The PCM looks at engine coolant temp, engine speed and A/C system pressure. If good, the PCM grounds the control circuit for the compressor relay.
Bottom line. If you have power from the IGN fuse to the relay and you are certain the relay is good, there is either a wiring problem between the fuse box and the PCM, or an issue with the PCM. Unless the BCM and HVAC Module were mucked with during the swap, the problem is most likely somewhere in the engine compartment. Either a high system presssure or a low system pressure will prevent the compressor from operating.