Suggest, you visit these hospitals and have a meeting with a doctor to get first hand information on your queries rather than depend on views of people at QL.
As for antepartum care, there are several private clinics in Doha which have qualified doctors on their panel. Naseem Al Rabeah Medical Centre located on C Ring road opposite Gulf Times is one of them. Aster Hospital is another. Doctors at Naseem Al Rabeah and at Aster could also answer all your queries. I guess Aster Hospital also has maternity facilities and accepts insurance. You could check up on that by calling them .
All the best!