That was a soul-crushing moment for Kate, she was basically cornered at that point, like being in check at a game of chess and forced to move your king out of harm's way just like your opponent wants, because he's a few steps ahead. Anthony could've ordered all the brainwashed kids to lynch her right there if she said no, so understandably she told him what he wanted to hear.
Douglas comes in clutch just as Kate felt helpless thinking she had no allies, but this is one situation I expected a few chapters before Anthony's lengthy backstory: so Kate is back to square one, but she's not alone like she thinks. I'm pretty sure there are others aside from Douglas that can help. Now would be a very interesting time for Maryrose and Rosemary to appear too.