The TIBC is quite high in spite of the high serum iron and saturation. This means your body is still trying to absorb more iron, even though there is already more than it needs. The high serum iron shows iron loading. Yet, your Hb does not rise even with a good supply of iron. This does look like thalassemia, but the HbA2 is below normal, and this is found in alpha thalassemia. Those with beta trait almost always have an HbA2 of over 5. Your Hb level is low for any type of minor. I think a DNA analysis looking for both alpha and beta thal should be done. In many ways, this looks like HbH disease, the 3 gene affected alpha thalassemia. The Hb is in the right range, as is the HbA2, and iron loading from diet can occur with HbH. But the MCH doesn't fit for HbH.
Have you taken iron supplements long term?