http://www.neorice.com/hoh_2337
Here you see another mistake she made.
First, take a look at the last two panels. That's not her mistake, it's just to establish some facts. Reichfelt betrays he knows about Piper's entourage, and that makes Piper notice he wasn't supposed to know that by himself, proving that he's complicit in the conspiracy. Because Piper hadn't told him about Pablo and Burk before.
Now that this is covered, move back to the first two panels.
Reichfelt talks about "spreading unfounded nonsense" and this doesn't cause the reaction that happens in the last two panels. Therefore, Piper
has told him before about the content of the message she wanted to send through the lighthouse. Despite knowing that her cousin may have been in on the conspiracy.
From that we can basically reconstruct what happened: Piper arrives, announces she wants to use the lighthouse, she's asked why, she doesn't make up an innocuous-looking lie as the reason but instead reveals that She Knows Too Much, Reichfelt gives the order to try to grab her staff, and as a result Piper must now
BURN EVERYTHING, which was presumably her intent in the first place.