Averaging exam scores with very different means/distributions

Follow what you defined previously. The 'average policy' needs to be done beforehand, so nobody can argue that you defined/changed it to specifically screw/help someone. If you didn't define it previously, the safer route is the arithmetic average, which is kinda standard (or something equivalent for your institution, if different).

More than that, IMHO, you are overthinking. Grades aren't a reward that you give, they are a measure of how much the student learnt the content. A professor of mine says that he doesn't give grades, the students earn it. If they performed badly, they performed badly. You have a duty to fail under performing students.