The proliferation of distribution-level green electricity and hydrogen resources entails an efficient local energy market (LEM). However, the existing LEM designed for electricity-hydrogen trading falls short of modeling multi-level mechanisms and accounting for the carbon intensity of hydrogen production. To bridge this gap, we propose a carbon-aware multi-level LEM for electricity-hydrogen trading based on a distributionally robust game framework, where hydrogen-based microgrids (HMGs) supply hydrogen to heterogeneous hydrogen users (HUs) including hydrogen refueling stations and industrial users. In this game framework, the coordination between HMGs and HUs is cast as a multi-leader multi-follower Stackelberg game. Specifically, HMGs determine an integrated hydrogen-carbon price, and carry out electricity trading through a non-cooperative game. Meanwhile, HUs act as followers, adjusting hydrogen purchasing strategies. Furthermore, the self-dispatching of HMGs and HUs is modeled as distributionally robust optimization problems considering source-load and hydrogen demand uncertainties, respectively. To hedge against these uncertainties, a novel Bayesian nonparametric hybrid ambiguity set is constructed based on local Wasserstein balls and moment information. Finally, the equilibrium of the proposed game framework is theoretically proved, and a distributed algorithm is developed to obtain this equilibrium. Comparative studies validate that the proposed game framework outperforms the existing ones, demonstrating a total income increasement of 12.3% and a carbon emission reduction of 11.6%.