Read up some proper peer reviewed studies on female endurance runners and birth control- there is no performance decrease on average. Also, oral contraceptive does not affect iron levels except to regulate periods to, in your case, decrease blood loss and therefore decrease the risk of anemia.
As a former D1 college runner myself, I recommend the combined pill, not the progesterone-only pills just for practicality of not stressing about taking it every day at the exact same time. I found there was none of the weight fluctuations that are so feared and actually made my iron levels go to normal for the first time in six+ years.
Also, read up on RED-S, the worse your menstrual function is, the worse you will perform. Oral birth control helps regulate periods and estrogen production, and can presumably help you avoid RED-S symptoms (including decreased bone density etc.). If you want to be healthy, running, or even functional as a 50+ year old woman in the future, consider this.
Good luck, make sure to do your own proper research on pubmed/ncbi before listening to these bozos.
-future sports doc, former runner