You need to have permanant Visa to visit Hamad.
If you are not able to obtain permanant visa, you'll have to visit the private clinics.
With respect to cost, Yes, the private clinics are quite expensive. But since you have no other options and since you are on visit, you have to visit the private clinics.

Aster Medcare in Hilal has one of the best docs Dr. Sabitha. From my personal experience is friendly and nice lady, answers almost all the stupid questions that me and my wife asked and it was surprising to see her very happy to see the test positive.
Consultation fee is QAR150 and any follow up within 10 days is free of charge.
U/s Scan - QAR300
Other tests are nominal price - provided you dont have a high-risk pregnancy where u need a detailed ivestigation.

Other private clinics could be around the same rate. And for deliver private hospitals cost you a forune.

If you manage to obtain ur PR, then you'll be saving a lot cos at Hamad is very minimal... no consultation fee, no fee for tests, QAR50 for opening a chart and QAR50 for u/s (which you'll be allowed to do only at 20 weeks)
and for deliver - QAR 500 max if normal. How cool is that...