I don't think Tesla advertises any P85D's with Ludicrous at all anymore, so no danger there. If they have an AP1 car advertised as AP1 car, but the previous owner replaced the side marker lights with AP2 version with cameras and you spot that in the CPO pictures, it doesn't mean Tesla is advertising the car as AP2. CPO pictures vary in quality, as I mentioned before, some have reported pictures with damages tires and wheels, so an incorrect badge is not out of the question.
That said, fake badges is probably less of a problem than ludicrous cars without the underline on the back or no badge at all. If on the other hand you pay attention to the badge on the instrument cluster or about screen as I suggested, seeing an underline there is a safe bet the car has ludicrous, and not seeing it pretty much means it doesn't have it - it is a lot more reliable than the rear badge.