It doesn't sound like the agent was mistaken about policy, just about you

I've had my BP (with AC*G on it) scanned at a few United Clubs, then had the agent ask something along the lines of "how are you getting in today?" "Air Canada Star Gold" has always been a sufficient answer if my BP says Star Gold.

Handing over my card with my BP avoids the question in the first place.

Basically, don't even try to get into a lounge without presenting your card (if you have the card). The time you save by not taking it out of your wallet is going to be a lot less than the extra time they'll need to check you in.