None of your three suggestions is quite right.
1. Aviation exam - no, because aviation is something different.
2. Aeronautical exam - no, because this is an adjective describing a quality. The exam itself isn't aeronautical.
2. Aeronautic exam - no, because 'aeronautics' is plural.
So, what do you call an exam in aeronautics? Quite simply, you call it 'an aeronautics exam'.
A mathematics exam
An economics exam
An aeronautics exam
These are compound nouns. The second word tells you what it is (an exam) and the first word tells you what kind of exam it is - an aeronautics exam.