Offshore aquaculture fish farming faces labor shortage, safety, productivity and high operating cost issues. Unmanned underwater vehicles (UUVs) are being deployed to mitigate these issues. One of their applications is the fish net-pen visual inspection. This paper aims to develop and simulate with high-fidelity several trajectory tracking control schemes for a UUV to visually inspect a fish net-pen in a standard task scenario in offshore aquaculture under 0.0 m/s, 0.5 m/s and 0.9 m/s underwater current disturbances. Three controllers, namely 1) Proportional-Derivative control with restoring force & moment compensation (Compensated-PD), 2) Proportional-Integral-Derivative control with restoring force & moment compensation (Compensated-PID), and 3) computed torque (or) inverse dynamics control (CTC/IDC) were conducted on a 6 degrees-of-freedom (DoF) BlueROV2 Heavy Configuration dealing with 12 error states (pose and twist). A standard task scenario for the controllers was formulated based on the Blue Endeavour project of the New Zealand King Salmon company located 5 kilometres due north of Cape Lambert, in northern Marlborough. This simulated experimental study gathered and applied many available and physically quantifiable parameters of the fish farm and a UUV called BlueROV2 Heavy Configuration. Results show that while utilizing the minimum thrust, CTC/IDC outperforms Compensated-PID and Compensated-PD in overall trajectory tracking under different underwater current disturbances. Numerical results measured with root-mean-square-error (RMSE), mean-absolute-error (MAE) and root-sum-squared (RSS) are reported for comparison, and simulation results in the form of histograms, bar charts, plots, and video recordings are provided. Future work will explore into advanced controllers, with a specific emphasis on energy-optimal control schemes, accompanied by comprehensive stability and robustness analyses applied to linear and nonlinear UUV models.