This Monte Carlo simulation examines how prompt engineering strategies shape the quality of large language model (LLM)--generated personality assessment items within the AI-GENIE framework for generative psychometrics. Item pools targeting the Big Five traits were generated using multiple prompting designs (zero-shot, few-shot, persona-based, and adaptive), model temperatures, and LLMs, then evaluated and reduced using network psychometric methods. Across all conditions, AI-GENIE reliably improved structural validity following reduction, with the magnitude of its incremental contribution inversely related to the quality of the incoming item pool. Prompt design exerted a substantial influence on both pre- and post-reduction item quality. Adaptive prompting consistently outperformed non-adaptive strategies by sharply reducing semantic redundancy, elevating pre-reduction structural validity, and preserving substantially larger item pool, particularly when paired with newer, higher-capacity models. These gains were robust across temperature settings for most models, indicating that adaptive prompting mitigates common trade-offs between creativity and psychometric coherence. An exception was observed for the GPT-4o model at high temperatures, suggesting model-specific sensitivity to adaptive constraints at elevated stochasticity. Overall, the findings demonstrate that adaptive prompting is the strongest approach in this context, and that its benefits scale with model capability, motivating continued investigation of model--prompt interactions in generative psychometric pipelines.