Insurance (and your doctor) likely won't approve infusions INSTEAD of supplements. The infusions are to boost levels more quickly than the supplements. You keep taking the pills *while* getting the infusions (and after).
I had no side effects from infusions except a little fatigue, just to balance a PP's experience.
Get an actual iron panel (bloodwork) so you know what your numbers are.
Also, "birth control" is not a single thing. There are dozens of brands of oral contraceptives, and they can have very different effects (in terms of mood, skin, flow, weight gain/loss...)