Well 22 is ludicrously low. You need to get up to 80. Ferrochel might be worth a go, look at Solgar. But my instinct is that maybe there's an underlying problem - celiac? ever had a test? Very cheap on line. H pylori - ever tested for that? If you tell your GP you have reflux s/he should agree to order an h pylori stool test. I read - and you can easily check with a bit of Google fishing - that 35 per cent of those with Hashi's also have autoimmune gastritis (AIG) - which can lead to low iron. It can also lead to low B12, and to pernicious anemia, but we're all mainlining supplements these days so my guess is those symptoms could be well disguised and it might just show as a malabsorption problem. Have a look on Google at that condition too and see what you think. I'm about to test privately for AIG and I think once you've discounted the first two you might think about that.
Do you drink loads of milk? Because that can reduce iron absorption.
Have you tried gluten free (after a celiac test)?
Any chance you are lactose intolerant?
There's something going on. Very heavy periods could do it. But malabsorption could too.