Beyond-diagonal reconfigurable intelligent surface (BD-RIS) has recently been introduced to enable advanced control over electromagnetic waves to further increase the benefits of traditional RIS in enhancing signal quality and improving spectral and energy efficiency for next-generation wireless networks. A significant issue in designing and deploying BD-RIS is the tradeoff between its performance and circuit complexity. Despite some efforts in exploring optimal architectures with the lowest circuit complexities for ideal BD-RIS, architecture discovery for non-ideal BD-RIS remains uninvestigated. Therefore, how non-idealities and circuit complexity jointly affect the performance of BD-RIS remains unclear, making it difficult to achieve the performance - circuit complexity tradeoff in the presence of non-idealities. Essentially, architecture discovery for non-ideal BD-RIS faces challenges from both the computational complexity of global architecture search and the difficulty in achieving global optima. To tackle these challenges, we propose a learning-based two-tier architecture discovery framework (LTTADF) consisting of an architecture generator and a performance optimizer to jointly discover optimal architectures of non-ideal BD-RIS given specific circuit complexities, which can effectively explore over a large architecture space while avoiding getting trapped in poor local optima and thus achieving near-optimal solutions for the performance optimization. Numerical results provide valuable insights for deploying non-ideal BD-RIS considering the performance - circuit complexity tradeoff.