Integrated Satellite Aerial Terrestrial Networks (ISATNs) are envisioned as key enablers of 6G, providing global connectivity for applications such as autonomous transportation, Industrial IoT, and disaster response. Their large-scale deployment, however, risks unsustainable energy use and carbon emissions. This work advances prior energy-aware studies by proposing a carbon-aware orchestration framework for ISATNs that leverages Digital Twin (DT) technology. The framework adopts grams of CO$_2$-equivalent per bit (gCO$_2$/bit) as a primary sustainability metric and implements a multi timescale Plan Do Check Act (PDCA) loop that combines day-ahead forecasting with real-time adaptive optimization. ISATN-specific control knobs, including carbon-aware handovers, UAV duty cycling, and renewable-aware edge placement, are exploited to reduce emissions. Simulation results with real carbon intensity data show up to 29\% lower gCO$_2$/bit than QoS-only orchestration, while improving renewable utilization and resilience under adverse events.