In this work, the symbol error rate (SER) performance of a relay-assisted underwater wireless optical communication (UWOC) system has been investigated over the composite exponential-generalized gamma (EGG) distribution with the beam spread function (BSF) under two hard decision schemes of fixed decision threshold (FDT) and dynamic decision threshold (DDT). Specifically, the oceanic turbulence is assumed to follow the EGG distribution, and the impacts of absorption, scattering and misalignment loss are characterized by BSF. The cumulative distribution function (CDF) of this UWOC system is derived with the max-min criterion as the best path selection scheme. And with the help of Gauss-Laguerre quadrature function, the analytical SER expressions for these two threshold schemes are then achieved and validated by Monte Carlo (MC) simulation. Moreover, the SER performance is further studied under different temperature gradients, bubble levels (BLs) and water salinity over three water types, as well as the system structure parameters. Results show that the UWOC system with DDT scheme can efficiently overcome the error floor induced by FDT scheme and demonstrates better SER performance. Furthermore, the SER performance would be improved with lower BL, temperature gradients and water salinity as well as the concentration of dissolved particles. This work will benefit the design and research of relay-assisted UWOC system.