Wireless Channel-based Secret Key Generation (WC-SKG) offers a promising alternative for wireless communication security, yet suffers from an extremely low key generation rate (KGR) in quasi-static environments (e.g., indoor Internet of Things), where key refresh cycles can extend for hours, creating a security vulnerability. Reconfigurable Intelligent Surfaces (RIS) can boost KGR by introducing artificial randomness, yet existing schemes only randomize the RIS phase shifts without using environmental priors, failing to maximize the KGR and meet the communication signal-to-noise ratio (SNR) requirement, causing service outages. To overcome these limitations, we propose the RIS-assisted Integrated Communication and WC-SKG (RICK) scheme. First, we treat the RIS-adjusted channel as a designable random variable and derive its optimal probability density function (PDF) using a novel geometric-algebraic framework to maximize the KGR under the communication SNR constraint. Second, we design a constrained-clustering-based quantization region division scheme tailored to this optimal non-uniform PDF, guaranteeing uniformly distributed secret keys. Simulation results show RICK achieves a KGR approximately 3.5-5 times higher than the state-of-the-art scheme while saving at least 20 dB of transmit power for the same KGR, confirming its effectiveness in quasi-static scenarios.