Beyond diagonal reconfigurable intelligent surfaces (BD-RIS) have emerged as a promising technology for 6G wireless networks, offering more advanced control over electromagnetic wave propagation than conventional diagonal RIS. This paper proposes a novel integrated sensing and communication (ISAC) framework that incorporates BD-RIS at the transmitter. This not only opens the door to enhanced sensing and communication performance, but also alleviates the need for large-scale fully digital radio frequency (RF) chains at the transmitter. Based on the proposed system model, we formulate a normalized weighted optimization problem to jointly design the active beamforming and the BD-RIS scattering matrix with the aim of jointly minimizing the trace of the Cramér-Rao bound (CRB) for sensing targets and maximizing the sum rate (SR) for communication users. To address this highly coupled optimization problem, we propose a novel and low-complexity iterative algorithm that efficiently solves the active beamforming and scattering matrix subproblems by transforming each into a series of tractable projection problems with closed-form solutions. Numerical results show the appealing capability of the transmitter-side BD-RIS-aided ISAC over conventional diagonal RIS-aided ISAC in enhancing both sensing and communication performance. Moreover, compared to the classic iterative algorithm, the proposed algorithm offers enhanced dual-functional performance while significantly reducing the computational complexity.