People often say clinic and hospital interchangably here. Often when they say clini they mean the free government clinics -- which aren't bad but if you've got good insurance (which you do) head straight to a private hospital for any kind of treatment.

There's Doha Clinic, Al Ahli, Al Emadi, American Hospital. I can't swear which ones handle direct billing with your insurance (you might want to call ahead if that's a concern), but you should definitely be able to get reimbursed for a visit to any of them.