Maxperson
Morkus from Orkus
First, I wouldn't use an adventure module as proof of anything about the actual rules of the game, especially modules that were written early on in the design process.
Leaving that aside, the doppelganger can try to deceive the PCs. It just tells the lies and then it's on the players to declare an action along the lines of wanting to determine the doppelganger's truthfulness (or agenda or personal characteristics). At that point, the DM can - if he or she finds the player's stated action to have an uncertain outcome and a meaningful consequence of failure - use the doppelganger's Charisma (Deception) check to set the DC for the PC's Wisdom (Insight) check. If the PC succeeds, then the narrated result can be that the doppelganger is not being truthful. If the PC fails, then the narrated result can be that the PC can't tell if it's being truthful or not. In either case, it's up to the player how the character responds.
You could also give the PC disadvantage, since the doppleganger is supposed to have advantage. Just switch it around so it's the PC making the check, but it's a very difficult check.