Student missing exam for “medical reasons”
In my experience dealing with these types of things correctly takes more time and effort then letting them slide. Continue scheduling exam times for her until she takes the exam or the incomplete automatically converts to a fail. I would even reuse the last remediation exam instead of making a new exam. I would only schedule the exam at times that are convenient for me.
It seems you have the answer with regard to procedure. The question now is what you should do in terms of "fairness" or ethics.
Educators and students should approach the work from the same perspective - for the student to learn. Academics should not be punitive, nor should they be a race to the bottom between the "good kids" who do everything right and hand their work in on time and the "slackers" or "hot messes" who can't seem to get anything in on time (flanked by the excuse-makers who suffer heavily from the Dunning-Kruger effect). The most pleasurable students to teach are those who truly care about the subject matter. Unfortunately, a lot of students forget this and are instead overwhelmed by the stress of grades, lack of structure, etc.
Your best bet is to have a frank discussion with the student. Is she currently in danger of failing the class? Is she a borderline case? Whatever you do, don't assume that she's being manipulative. Instead, tell her that you would like to reschedule, but you're concerned that this might happen again. Ask her how she thinks she's doing in the class and what she would like to have happen. From there, talk options. Does she have the option to just drop the class (at most places, faculty can get around deadlines rather easily)? What is her worst case scenario? Is she prepared for the final? She should have the option to take the exam, but it might be a good opportunity to talk about actions she can take to prevent this sort of thing from happening in the future. Talk about learning from failure and taking measures to prevent this in the future. It's possible she's a pathological liar, but it's also possible that she's a kid who needs more guidance navigating the adult world.
No matter what, you ought to give her a chance to take the exam again and open communication about her status in your class. Invite her to your office hours to talk and tell her to email you back if she can't make it. If she fails to respond, then you have to fail her, but making sure that it's an emotionally neutral experience is probably your best bet.
I would allow rescheduling, but be sure to use a different exam.
As some other answers indicated, one probably does not want to get embroiled in the details of the student's medical situation, and it may be inappropriate, besides. (I think it would be illegal in the U.S. to ask for details beyond a general note from the doctor.) Thus, one simply cannot have the information to know whether missing the exam was frivolous or not.
With regard to communication, yes, in principle one imagines that the student would have been able to contact you in advance to explain that (s)he'd miss the exam. But, again, without prying, one cannot know.
Yes, the odds may be good that the situation is due to student negligence than medical emergencies... and we should hope so, in the larger scheme of things. But, since we cannot be sure, my choice would be to treat the situation as a medical emergency, whose details I will not know. And allow further exam retake.
As to whether allowing further exam retakes is fair to the students who've managed to do things on schedule... if there was a genuine medical emergency, we are attempting to compare incomparables. If not, my consolation is that, in my experience, students who miss exams due to their own negligence or disorganization similarly fail to "profit" by extra chances.
Thus, I take claims of medical/personal emergencies at face value, and do not restrict re-takes of exams.