A hemoglobin electrophoresis test (simple blood test) can tell you if it's beta thal. The HbA2 and HbF will test at higher than normal.
Iron should never be taken unless iron deficiency is diagnosed by a panel of test. Your son has no iron deficiency, so this should be avoided, as iron can build up and eventually damage the organs. Iron cannot help the anemia of thal minor and can only help if iron deficiency is also present.
TIBC 308
UIBC 202
Iron serum 106
Iron Sat 34
Ferritin serum 194
Those numbers are all normal range, even towards the high end.