MAAF president Jason Torpy’s out-of-left field remarks about a USAFA religious training scenario were previously offered for consideration.  Based on what the regulations actually say, the following is most consistent with the environment of religious respect and religious freedom in the military, specifically tailored to the USAFA cadet culture: 

Based on DoD and Air Force policies (noted in the original question), a military member, including a USAFA cadet, is permitted to wear a necklace of any kind so long as it is not visible.  Demanding that a cadet not wear a necklace that was not normally visible would be inconsistent with the regulations.  Demanding that a cadet not wear a necklace because it held a cross would demonstrate not only ignorance of the regulation, but also a possible inappropriate targeted treatment of religion by a superior officer.

In an average, benign situation, the “commander (or teacher)” in this scenario would be wrong.  The cadet should be allowed to wear the cross (in compliance with the regulations).

However…

Phoenix Blue’s comment on the last post wins the virtual Christmas turkey.  Recall the DoD policies and their exceptions:

…provided they shall not interfere with the performance of the member’s military duties…

Now read the scenario again:

Lisa wears a cross under her uniform. On the obstacle course one day…

Every USAFA cadet experiences the obstacle course at Jack’s Valley during basic training.  Cadets and officers go through problem-solving “leadership labs” that involve similar obstacle courses at various points in their careers.  In virtually all cases, they have the same regulation:  No jewelry.  There’s a simple, discrimination-free reason:  It’s a safety hazard.

So, in this case the “commander (or teacher)” may be entirely correct:  “Lisa” can’t wear the cross on the obstacle course because it violates safety regulations.

This scenario appears to actually be an excellent training tool, as it gives cadets the opportunity to discuss “competing” regulations and seeming conflicts between liberties and military duties — something they will see throughout their military careers.  The cadets can learn far more by examining all possible aspects of what could be than they would if they were simply trying to decipher a canned book answer.

It seems Jason Torpy could, too.