I don't think many with HbH transfuse regularly. The most important nutrient for alpha thals is folate. Iron isn't a good idea as a supplement, because when your Hb is low, your body will absorb more iron than needed.
You should be on folate. If your doctor disagrees, he is disagreeing with the single universal recommendation for HbH, folate. Folate is needed to build red blood cells, RBC's and unlike iron, it is quickly eliminated from the body because it is water soluble, so it needs to be replenished regularly. Your bone marrow is overactive, trying to compensate for the low Hb, and you need to provide folate to keep up with the demands of this process. And it should be a therapeutic dose of at east 2-5 mg daily.