Low Earth orbit (LEO) satellite communications (SATCOM) with ubiquitous global connectivity is deemed a pivotal catalyst in advancing wireless communication systems for 5G and beyond. LEO SATCOM excels in delivering versatile information services across expansive areas, facilitating both unicast and multicast transmissions via high-speed broadband capability. Nonetheless, given the broadband coverage of LEO SATCOM, traffic demand distribution within the service area is non-uniform, and the time/frequency/power resources available at LEO satellites remain significantly limited. Motivated by these challenges, we propose a rate-matching framework for non-orthogonal unicast and multicast (NOUM) transmission. Our approach aims to minimize the difference between offered rates and traffic demands for both unicast and multicast messages. By multiplexing unicast and multicast transmissions over the same radio resource, rate-splitting multiple access (RSMA) is employed to manage interference between unicast and multicast streams, as well as inter-user interference under imperfect channel state information at the LEO satellite. To address the formulated problems non-smoothness and non-convexity, the common rate is approximated using the LogSumExp technique. Thereafter, we represent the common rate portion as the ratio of the approximated function, converting the problem into an unconstrained form. A generalized power iteration (GPI)-based algorithm, coined GPI-RS-NOUM, is proposed upon this reformulation. Through comprehensive numerical analysis across diverse simulation setups, we demonstrate that the proposed framework outperforms various benchmarks for LEO SATCOM with uneven traffic demands.