How about THIS?!:
Aizawa lets the matter rest, but recovery girl can't get the 'palm hairs' out of her head! She checks the internet for 'barbed hairs in animals' and puts two and two together! Aizawa finds out too (Peter admits to the Oscorp part and how such knowledge is DANGEROUS in the WRONG hands. Aizawa agrees to keep it a SECRET), but instead of putting a stop to Peter's 'activities', he gets him a temporary license. Later, when Peter learns of Aizawa's and AM's PRIOR suspicions of him (along with AM's secret), the trio discuss on what kind of quirk the mole might have, with Peter speculating the quirk might not be anything that could be considered SPECTACULAR, and maybe the person's MENTAL profile, for all they know, is NOWHERE CLOSE to what AM has in mind (angry and vengeful. Instead, this time, the person could be a GULLIBLE individual, as a SMART bad guy isn't going to use the same trick again.), and they could be someone the staff is likely to ignore. Peter's knowledge on quirk-drawbacks helps them conclude the mole's new quirk may be something that hasn't been FULLY accepted by their body, as forcing a quirk into a body DEVOID of the quirk-gene can cause the quirk to hurt the user's body every time they use it, with Peter comparing the quirk to an organ (as in some instances, organs transplanted into bodies that do not share the same DNA are LIKELY to be rejected by the bodies, leading to the hosts' death. Even those who receive organs from relatives have to consume anti-rejection meds), so this time, they are nothing more than a pawn and spy relaying info to the enemy!