Self-report questionnaires remain the default tool for probing the psychological characteristics of Large Language Model (LLM) agents, yet classical instruments (BFI, BDI, MBTI, BSS) inherit three well-known threats under LLMs: contamination from training corpora, directional bias under social-desirability framing, and limited responsiveness to context beyond the item text. We ask whether a *projective* paradigm can be adapted into a usable psychometric tool for LLM agents. We introduce **GenPT** (Generative Projective Testing), which reformulates TAT, Rorschach, and SCT with newly generated stimuli and organises assessment as a three-stage pipeline (Behavior Collection → Interpretation → Diagnosis) grounded in SCORS-G and a Simplified Rorschach Analysis System. On personality traits (Big Five, MBTI) and mental-health risks (depression, suicide ideation), questionnaires exhibit systematic directional shifts under social-desirability framing, most strongly on suicide ideation, whereas GenPT’s collected behavioral patterns stay near the symmetric baseline; under a longitudinal counselling context, GenPT-based depression assessment shifts by roughly an order of magnitude more than its questionnaire counterpart. Questionnaires remain competitive on clean-persona trait tasks where items align lexically with the persona description. Overall, GenPT complements rather than replaces self-report when contamination resistance, bias asymmetry, and context sensitivity matter. Code and stimuli: https://github.com/sci-m-wang/GenPT.