I think it will be difficult to find a specialist like gastroenterologist privately. The easier route IMHO will be to go through the GP in one of the govt. health centers and take their recommendations for an specialist in Hamad. GPs in these health centers normally do this if they think you need to see an specialist and prepare you for that with all the necessary pre-examinations. So that your test data will already be there in their network when you see the specialist.