Lucille,
Looks like no one really knows the answer at this stage
I think your question has been answered. You cannot use an AVID for unescorted access at a security controlled airport where display of a valid ASIC is required.
One variation, not often exploited under Reg 3.03 is that the display of an ASIC only applies during the operational period at airports where screened services do not operate. Any screened services and it is a 24/7 requirement.
However you will always get the local airport staff demanding you wear an ASIC regardless, because their TSP says so. Which raises an interesting dilemma, does the TSP trump the legislation? How can a pilot be expected to know the contents of and airports TSP when it is not made available? But then the airport operator could simply deny you access which is their right to do also. And around and around we go.