Must be beginner's luck : I seem to have found a friendly and competent watchmaker who accepted to look at my watch
He did make a few comments about missing screws and cheap components but he will look into the problem and says it will cost anywhere between 20 and 120 euros depending on the cause of the problem.
I have to wait a few more days before declaring this a total success but I'll definitely keep you posted...
By the way, you could walk there from tram 33 but there's a bus that takes you closer