If one had to choose between the Sergeant Audie Murphy Club and a master's degree, which would you pick?

That's easy...masters degree

No reason not to do both.

Assuming that the Master's degree is relevant and not in underwater basket weaving, I'd vote for that. SAMC is indeed an accomplishment and a great selling point for promotions, but a Master's degree seems to carry much more weight both in the military and in the civilian sector. Remember, you never see a job posting requiring SAMC membership, but you see plenty that require a Master's degree.