I should know bette than to argue with with HoH ruling, but I must since I feel they got it wrong here. The rules define “identity” as “a player card type that represents which character a player is playing in the game. A player’s identity card is a double-sided card that represents their hero on one side and their alter-ego on the other. The side that is face up indicates the form (hero or alter-ego) that player is currently in. Each player begins the game in alter-ego form. If a card refers to a hero or alter-ego by title, it refers only to the identity with that title, and not to the other side of the card.”
Since this card requires your “identity” to have the avengers trait, this applies to both sides of your double-sided card, since it does not refer to any one form (alter-ego or hero). Thematically it makes more sense as well.. because why would Cap lose his authority to recruit an Honorary Avenger just because he and flipped to civilian side? I have a hard time imagining him saying “Sorry Star-Lord. I’d love to recruit you, but as you can see, I’ve already changed clothes so I no longer have the authority to make you that offer.”
Similarly, new characters with mutant traits don’t lose those traits based on what costume they are wearing…. They are still mutants because their identity (either side/form) says so.