Companies deploy large language model (LLM) assistants on their websites to advise customers. For commercial reasons, the operator can give an assistant a goal that the customer never sees. This paper focuses on operator-induced vulnerability-targeted manipulation (VTM), in which an assistant pursues such a goal by exploiting a vulnerability the customer discloses. While existing research mainly measures how manipulative a model is, we go further and ask whether VTM can be detected from the customer-AI conversation. To investigate this at scale, we introduce GAP (Goal-Anchored Probing). GAP first records the option a model recommends without any goal and then sets the other option as the operator's goal. We apply GAP to 15 open- and closed-source LLMs and collect 619,600 labeled replies across 540 everyday scenarios in 12 domains. We organize our study around three research questions. (1) Do assistants with an operator goal target vulnerable customers? (2) Can VTM be detected from the customer-AI conversation? (3) Does detection generalize to unseen models, vulnerabilities, and instruction styles? We first find that 14 of 15 models push the operator goal, and 12 of them use disclosed distress or financial strain to support it. Next, we show that fine-tuned detectors identify exploitative pushes with 79\% accuracy from the reply alone, while zero-shot LLM judges average at most 62\%. Finally, detection weakens on vulnerabilities absent from training. Telling whether a push comes from the operator also becomes unreliable when the assistant is told to hide its goal. We will release our dataset and code to facilitate future research on auditing LLM assistants.

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