The Compressor wont engage because its low on Gas,
there is a High and Low Pressure Switch that measures the amount of pressure in the system,
and if it too low,
it WILL NOT allow the compressor to engage,
so you don't damage the system hardware with little to no gas in the system.
With the experience over the years I have had with Air Con systems, also Car systems,
The first thing to do, is get the system pressure tested, that will reveal the result.
I would probably be replacing the Compressor, as I would bet that the Gas has leaked out via the Compressor,
as the Calais has a Climate Control system and it starts working when you start the car.
and again make sure they use Fluorescent oil/dye in it,
So you can see where its leaking from with a UV Torch.
I have a noisy compressor, it vibrates when the car is idling down,
So I have one here to replace it with I purchased from a wrecker. [$66]
Then vac it out, and gas it up.