The system is harvesting methane from Earth and converting it to carbon dioxide (another greenhouse gas, but less potent). If the system uses the Sabatier meth

The system is harvesting methane from Earth and converting it to carbon dioxide (another greenhouse gas, but less potent). If the system uses the Sabatier method to produce methane from CO2, then it's using solar power (I can't imagine doing it in any other way sanely) to convert CO2 to CH4, which is then converted back to CO2 by the rocket. Net gain of no greenhouse gas.