Why didn’t the mom research the law and see the rape exception?
Miss. Code § 41-41-45 requires a formal charge of rape by law enforcement for the exception to be applicable. If you read on to the Time article linked at the end of the Daily Beast article the mother claims to have been unaware of the rape to begin with. Even if charges could, as a matter of procedure, be filed against a John Doe to satisfy the requirement you still have to convince law enforcement to file those charges. Just informing a doctor that the pregnancy was the result of rape is insufficient to satisfy the exception.
Now you might be thinking, "Well, Mom didn't know but the kid knew what happened to her." I point to this from the Time article:
Regina hadn’t yet explained to her daughter how a baby is made, because she didn’t think Ashley was old enough to understand. “They need to be kids,” Regina says. She doesn’t think Ashley even realized that what happened to her could lead to a pregnancy.
It is unsurprising that she might lack the understanding and foresight required to be her own advocate on this issue.
I would hope that people would take the time to investigate a topic that was relevant and so important but one of the problems with ignorance is that sometimes you are so ignorant you don't know you have a blind spot. If you've not paid attention beyond hearing that abortion is now banned the thought, "I wonder if there is a rape exception?" may just not occur. A bit like how so many people don't know how tax brackets work, assume they do, and never bother to look into the topic.
As far as the doctors. I looked into things a bit more and apparently Mississippi's last abortion clinic closed down in July of 2022, which isn't surprising given the legal situation, so the direction to Chicago by the doctor may not have been motivated by advice to avoid persecution but rather it being the nearest qualified facility. So I think we were both erroneously assuming there were in-state options that weren't being used.
P.S. I suspect Chicago wasn't literally the closest but probably the closest without restrictions such as requiring a separate counseling and procedure visit that creates a greater barrier than a longer drive.