Conventionally speaking, yeah, that's usually how it works. The PCM looks at the pressure switch to determine if there's sufficient system pressures, and grounds a relay to provide power to the compressor clutch.
Your 3.xx volts at the clutch concerns me though. It should have to be around battery voltage for the thing to engage. Was that 3.something with the connector to the compressor clutch disconnected, or by backprobing the connector?
Somewhere, I think you're losing about 9V. I'd start with checking grounds - leads coming off the battery, grounds coming off/going to the block, the battery cable ends themselves, make sure all the various grounds you see are tight and clean. I would think it should have a separately serviceable relay somewhere, the clutch draws a lot of juice on initial start up. Check your powers and grounds at the relay, or if it shares a similar relay design in the car, swap it with that and try again.