Conversational Recommender Systems (CRSs) engage users in multi-turn interactions to deliver personalized recommendations. The emergence of large language models (LLMs) further enhances these systems by enabling more natural and dynamic user interactions. However, a key challenge remains in understanding how personality traits shape conversational recommendation outcomes. Psychological evidence highlights the influence of personality traits on user interaction behaviors. To address this, we introduce an LLM-based personality-aware user simulation for CRSs (PerCRS). The user agent induces customizable personality traits and preferences, while the system agent possesses the persuasion capability to simulate realistic interaction in CRSs. We incorporate multi-aspect evaluation to ensure robustness and conduct extensive analysis from both user and system perspectives. Experimental results demonstrate that state-of-the-art LLMs can effectively generate diverse user responses aligned with specified personality traits, thereby prompting CRSs to dynamically adjust their recommendation strategies. Our experimental analysis offers empirical insights into the impact of personality traits on the outcomes of conversational recommender systems.