Future wireless networks, in particular, 5G and beyond, are anticipated to deploy dense Low Earth Orbit (LEO) satellites to provide global coverage and broadband connectivity. However, the limited frequency band and the coexistence of multiple constellations bring new challenges for interference management. In this paper, we propose a robust multilayer interference management scheme for spectrum sharing in heterogeneous satellite networks with statistical channel state information (CSI) at the transmitter (CSIT) and receivers (CSIR). In the proposed scheme, Rate-Splitting Multiple Access (RSMA), as a general and powerful framework for interference management and multiple access strategies, is implemented distributedly at GEO and LEO satellites, coined Distributed-RSMA (D-RSMA). By doing so, D-RSMA aims to mitigate the interference and boost the user fairness of the overall multilayer satellite system. Specifically, we study the problem of jointly optimizing the GEO/LEO precoders and message splits to maximize the minimum rate among User Terminals (UTs) subject to a transmit power constraint at all satellites. A robust algorithm is proposed to solve the original non-convex optimization problem. Numerical results demonstrate the effectiveness and robustness towards network load and CSI uncertainty of our proposed D-RSMA scheme. Benefiting from the interference management capability, D-RSMA provides significant max-min fairness performance gains compared to several benchmark schemes.