If you're not getting 12V at the clutch to engage it via the "proper" system settings, then I suspect it is either
1) an A/C relay issue (relay is not closing, and therefore the required current at 12V to engage the A/C clutch is not getting through)
2) a wiring issue between the relay and the pressure switch. Jumpering the pressure switch will engage the AC clutch only if the current is there to flow through the bypass. This is true regardless of charge level AFAIK. The fact that the clutch doesn't engage indicates likely there is no current flowing to the clutch, and since jumpering the switch didn't work, that implies that the switch is not seeing 12V either. So the issue is upstream.
3) There is also the chance that the "AC on" command from the BCM is either not being sent to the input of the relay -- this would be a BCM issue--- OR the wiring to/at the relay on the underside of the fuse block may be damaged/contacts melted, etc,
4) wiring issue between BCM and AC relay.
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Since you indicated that the system does not seem to release refrigerant when you depress the valve, you are likely low on refrigerant or completely out.
When this happens, I jumper the pressure switch long enough for the suction line to pull in refrigerant to exceed the low pressure cut-out.
Since you can't do this, you will need to manually apply the 12 V to the clutch as you did when you tested it. I would leave the pressure switch UNJUMPERED while doing this. Every so often, CLOSE and detach the refrigerant can and check the low side port with a gauge--with the 12V applied so the clutch is still being manually engaged. Get the system about half charged. You will have then eliminated low system pressure from the list of possible causes.
Remove the manually applied 12V at the clutch.
I expect the clutch will disengage.
Jumper the pressure switch.
.If the AC clutch kicks in, the pressure switch is bad.
If the A/C clutch does not kick in at this point, I would say AC relay or wiring which can be traced to where the 12V current feed to the circuit disappears as discussed above