I'm very curious as to how your buildings AC guy assessed this system without it running or him removing and weighing and reloading the correct weight of R134A back in. I'd say that all he did was to check its standing pressure, and that will always be the same regardless of how much gas is in there as long as it is enough to keep the pressure at the correct pressure for the ambient temperature. From my limited experience of car AC systems and building Fridge people, they will not touch car systems with a barge pole, which is fair enough, most if not all of them will be running company leased cars that have service support built in, and if the AC is not working, they will get it booked in to get fixed, a bit like horses for courses.
So, where does that leave you, well it sounds like there is some R134A in that system still, so I'd think that the compressor should be able to get enabled when demanded - and that the set of conditions that must be satisfied to allow the compressor to start to run are not being met. If it is short of gas, the compressor would run initially for a short period before shutting down, if the gas was short but only slightly short, the system would also run but not cool down far enough.
I think that this car needs investigating with a proper VW Group full compliant scan tool to try to work out what is wrong, and that might start by checking that the controller is okay, followed by is the compressor control lines being complete. Also is that car old enough to have a clutch controlled compressor drive or is it a variable displacement always driven version of compressor. The pressure sensor is always a possible failed component but I'd only replace that with a genuine VW Group one and they cost quite a bit so no point in just swopping parts to try to get this system working again, using diagnostics should be the way forward followed by good old "getting stuck in " and physically checking wiring ect.
Edit:- I'm in the opposite position to you, I just need a dummy place like Halfords/ATS etc to reload the system in my wife's 2015 Polo so that I can try to find running leaks or just live with it losing gas very very slowly and get the AC back running as currently it is too low on gas charge and I don't feel like wasting my small stock of R134A only for it to escape.