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Welcome to the forum.
You are correct that the A/C system monitors the gas pressure in the system and will not engage the compressor clutch if the pressure is too low. This appears to be what has happened in your case.
A/C systems rely on the gas circulating to lubricate the moving bits of the compressor and keep the seals intact. That is why it is important to run the A/C regularly even in winter otherwise seals dry out and end up leaking and bearings dry out and squeak or seize... Expensive!
I suspect since you have not had the car long the system has been empty for a while and when you refilled it you are left with the symptoms of a dormant system.
When you took it to BMW they should have carried out a pressure test using nitrogen first as the system was empty. It is illegal to knowingly release refridgerant into the atmosphere so if there is a suspected leak they use nitrogen to determine where it is. They put the dye in ok but it sounds as if they did not spend enough time checking for leaks.
The usual place for a leak is in the condenser (the small radiator mounted in front of the engine coolant radiator) as these often get holed by stones being flicked up from the road but in your case it may be that the compressor seals have gone (squeakiness).
If the gas charge only lasted a week you have quite a serious leak and it should not be difficult for someone who knows what they are looking at to find. I would go back to the dealer and suggest to them that they did not adequately check the integrity of the system before filling it and you want them to identify where the gas is escaping from. They should do this for free since you have paid them to do that already! Once you know how the gas is getting out you can decide what to do but prepare yourself for a new condenser or compressor (perhaps just seals if available). You would need to pay for those to be fitted but if you get BMW to do the job (likely an arm and a leg!) you should at least get the refill gas for free.
Good luck! Let us know how it goes.
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