A
ferritin of 4 ng/mL is severely low, well below the normal range. That alone can explain the hair fall you're seeing. H. pylori infection often reduces how well the stomach absorbs iron, so the two problems are likely connected.
Next Steps
At a
ferritin this low, an IV iron infusion usually corrects the deficiency faster than oral tablets. H. pylori can further reduce how well oral iron is absorbed, so oral alone may take longer to work right now. Since your two doctors disagree, ask your treating doctor to decide based on your full case.
Health Tips
Get the H. pylori treated fully too, since it can keep draining your iron even after this correction.