On second thought, I have an alternative solution to what it is lampshading, that is the broken suspension of disbelief that after stating that the terms of Quirrell's contract prevented him and others from investigating Quirrell's identity, Albus would leave the room, allowing his conspirators to investigate Quirrell's identity.

This theory sounds incredibly plausible from the perspective of the author wanting to use the lampshade trope, but from the perspective of the reader, that action was completely in-character for Dumbledore and doesn't actually break suspension of disbelief.