With the goal of ubiquitous global connectivity, multibeam low Earth orbit (LEO) satellite communication (SATCOM) has attracted significant attention in recent years. The traffic demands of users are heterogeneous within the broad coverage of SATCOM due to different geological conditions and user distributions. Motivated by this, this paper proposes a novel rate-matching (RM) framework based on rate-splitting multiple access (RSMA) that minimizes the difference between the traffic demands and offered rates while simultaneously minimizing transmit power for power-hungry satellite payloads. Moreover, channel phase perturbations arising from channel estimation and feedback errors are considered to capture realistic multibeam LEO SATCOM scenarios. To tackle the non-convexity of the RSMA-based RM problem under phase perturbations, we convert it into a tractable convex form via the successive convex approximation method and present an efficient algorithm to solve the RM problem. Through the extensive numerical analysis across various traffic demand distribution and channel state information accuracy at LEO satellites, we demonstrate that RSMA flexibly allocates the power between common and private streams according to different traffic patterns across beams, thereby efficiently satisfying users non-uniform traffic demands. In particular, the use of common messages plays a vital role in overcoming the limited spatial dimension available at LEO satellites, enabling it to manage inter- and intra-beam interference effectively in the presence of phase perturbation.