Unmanned aerial vehicle (UAV) has become ideal platform for integrated sensing and communication (ISAC) due to its low-cost deployment and strong line-of-sight capabilities, meeting the demands for high-quality communication and precise sensing in 6G wireless networks. However, the beamforming design in UAV-assisted ISAC system suffers due to the half-duplex (HD) communication mode, leading to reduced system efficiency. To address this issue, this paper studies a UAV-assisted full-duplex (FD) ISAC system. This system utilizes a dual FD communication and radar transceiver architecture at the UAV, not only allowing it to reuse the same time and frequency resources for communication with both uplink (UL) and downlink (DL) users but also enabling it to detect target and process echo signal, aiming to maximize the transmission sum rate for both UL and DL users. Furthermore, a joint transceiver beamforming design algorithm is proposed. Initially, an equivalent communication model for sensing SINR is derived to establish its rate expression. Subsequently, an augmented weighted minimum mean squared error method is used to iteratively solve the problem through alternating iterations. Simulation results demonstrate that, compared to traditional HD ISAC, FD ISAC can effectively enhance the rate performance.