CAP is partially funded by the USAF but every document that comes out of CAP clearly and unambiguously states it is a civilian, 501(c) organization and I quote from the official declaration:
Civil Air Patrol is a congressionally chartered, federally supported non-profit corporation that serves as the official civilian auxiliary of the United States Air Force. CAP is a volunteer organization with an aviation-minded membership that includes members from all backgrounds
Please note the word "civilian". In fact CAP cadets are reminded, when in uniform, to salute anyone in uniform, regardless of rank, regardless of which of the US militaries & Coast Guard and Medical Service.
1) CAP members are expected to render military courtesy and salute all officers higher in rank, whether they be CAP, USAF, or other Military Branches. Active, Reserve and Guard component members are NOT required to render courtesies to us, but the are ALLOWED to (and when they do, it is appreciated, and returned).
2) According to Air Force instruction, CAP is considered part of the Total Force when operating on Air Force assigned missions, and are considered Airmen, alongside Active, Reserve, Guard and Civilian Components of the USAF.
In other words, even the highest ranking CAP officer is less than dirt when it comes to reality.
CAP senior members are frequently reminded (when something like the OP described) that CAP ranks, outside of CAP, mean absolutely nothing, they're the equivalent of Boy Scout merit badges. CAP members do NOT have on base privs unless they are offered as a courtesty (e.g. cadets during encampment at USAFA in the summer may use the on-base convenience store (the one NOT inside the cadet area) but may not use the Commissary.