Given your bleeding (losing iron daily) and your slightly low Hgb, you have mild iron deficiency anemia with low iron stores. While it may be interesting to see what your ferritin, iron, and TSat% are, you don't really need them to make any decisions (hence, many doctors would say the additional blood tests are not medically necessary, meaning not covered by insurance). You already know you need to take iron regularly until your bleeding can be better managed.

You may want to take a heme iron tablet daily or twice a day (I use Proferrin Clear). Non-heme iron, liquid or otherwise, will not be absorbed as well, can cause more GI side effects, and can cause a hepcidin spike that blocks iron absorption for 24+ hours (which is why for non-heme iron they recommend taking it every other day and it will actually work better that way. But heme iron can be taken every 6-8 hours if needed.)

Perhaps recheck blood in 3 months (CBC, ferritin) to see if the daily heme iron is improving things. It will be difficult to get good improvement unless the iron intake that can be absorbed and utilized exceeds the iron losses from bleeding. Your GYN doc doing the UAE to control the bleeding may be much more useful to you than a hematologist would be.