Reconfigurable intelligent surface (RIS) is a disruptive technology to enhance the performance of physical-layer key generation (PKG) thanks to its ability to smartly customize the radio environments. Existing RIS-assisted PKG methods are mainly based on the idealistic assumption of an independent and identically distributed (i.i.d.) channel model at both the base station (BS) and the RIS. However, the i.i.d. model is inaccurate for a typical RIS in an isotropic scattering environment and neglecting the existence of channel spatial correlation would possibly degrade the PKG performance. In this paper, we establish a general spatially correlated channel model and propose a new channel probing framework based on the transmit and the reflective beamforming. We derive a closed-form key generation rate (KGR) expression and formulate an optimization problem, which is solved by using the low-complexity Block Successive Upper-bound Minimization (BSUM) with Mirror-Prox method. Simulation results show that compared to the existing methods based on the i.i.d. fading model, our proposed method achieves about $5$ dB transmit power gain when the spacing between two neighboring RIS elements is a quarter of the wavelength. Also, the KGR increases significantly with the number of RIS elements while that increases marginally with the number of BS antennas.