I'm due in May also, and am in the same situation - trying to decide on the venue. I did not know that Morges was so popular that one may get turned down, that's interesting to hear and something I will now take in to consideration. As this is my first I'm not sure that I'd cope well in such a situation!
My Gyn/Op has suggested CHUV, naturally, as it is the main university hospital - but I still like the idea of a smaller more intimate atmosphere...and am also considering Vevey. Have any of you done tours of the ones mentioned?