Many of the answers so far have focused on either the fact that the student did this to himself (by taking too hard of a classload) or that exhaustion is not a valid excuse because it is common at the end of the semester.
This answer does not focus on the medical issue per se or its validity, but I wanted to point out that there are valid opposite side to these two issues.
Point 1. The student failed to manage his own time by signing up for so many classes.
I agree that some students fail in this regard, but these failures, around exam time in particular, are not always the student's fault. Example: when I was an undergrad, I majored in both math and molecular biology. My university scheduled exams over a 6 day period with exams for classes not commonly taken together typically scheduled on the same day. This is designed to accommodate most students who have picked one major, but for those of us who had interdisciplinary interests, the result was that we often had as many as 3 or 4 difficult exams crammed into something like an 18 hour window. For a junior or senior taking advanced classes in which the professor believes they know what other classes their students are taking and how difficult they are (because most students follow their department's curriculum), these conflicts can potentially be devastating. Ultimately, It may not be fair to allow two students different timelines for an exam in the context of the class, but it is also quite unfair for the algorithm that schedules exams to systematically disadvantage students who have interdisciplinary interests.
Point 2. Exhaustion is not a valid excuse, particularly because everyone is exhausted at the end of the semester.
I feel that this sounds convincing on its face, but it makes me uncomfortable. This line of thinking may come from a desire not to give anyone preferential treatment, but the net effect of this as a policy is that students with more resources get preferential treatment on average. Those students whose tuition and living expenses are covered by wealthy parents or who have the option of painlessly extending their degree to 5 years will always benefit from this policy on average while students who must work/raise children during college or who can't afford to spend more than 4 years getting their degree will always be hurt on average by this policy. One can argue that this is the best bias we can extract from a general policy, but I don't think one can argue that this policy is universally fair.
To be clear, I'm not arguing that one should or should not follow such a policy universally; rather I'm arguing that the policy has a non-trivial bias. The OP should consider if the biases implied by such a policy are biases that they are okay with in the context of their class/grades or whether they would prefer the biases carried by other policies.
Conclusion. I don't have a direct answer to the question, but I wanted to point out that rigid adherence to a policy position is not more fair just because it is more rigid. In particular, if the OP's goal, in teaching the class, is for the students' grades to best reflect their level of understanding of the topic (rather than to best reflect a standardized test on the topic) independent of other students, then adherence to a rigid attendance policy is probably antithetical to fairness. If the OP's goal is to have a clear paper-trail that shows they did nothing amiss, then the best thing is always to route things through the proper administrative channels. If the OP's goal is to prevent a situation where any other student might feel cheated, then obviously they should disallow the makeup exam. But if the OP's goal is to be as fair as possible to the student in question, then the answer isn't going to be found verbatim in a policy document.