This problem is common with all the commercial rail to rail output single power supply OPAMPs and it becomes worst with higher load currents. Only way to cover GND level with full BW of the loop is to use negative power rail to cover GND where loop will not be broken. Since a current sink could be used (Class A output) and this current sink would provide a good PSRR- than a Charge Pump type negative power suply (has ripple) could be used provided that your output current drive capability is small enough to keep chip area reasonable small. Good luck.