This causes one of the weakest points of the Earth's magnetic field, and thus leads to an increased flux of energetic particles in this region.
No, the particles come closer to Earth's atmosphere here due to the depressed magnetic field, not the other way around.
If a CME (Coronal Mass Ejection) from the sun were to hit this side of the Earth would the effects be greater than normal? By how much?
CMEs hit the Earth's magnetosphere when the western hemisphere of Earth faces the sun all the time, it's not a special event. The south Atlantic anomaly occurs because of the offset and tilt of the dipole moment of Earth's magnetic field from the rotation axis, e.g., see the following illustration (found at https://en.wikipedia.org/wiki/File:South_Atlantic_Anomaly.svg):
Regardless, most CMEs (i.e., all except the strongest) have little or no effect on the inner radiation belt while the outer belt can completely disappear and/or split into two belts under some conditions. During strong geomagnetic storms, yes the energetic particle fluxes in the inner belt are enhanced and thus, the fluxes in the south Atlantic anomaly are as well.
Would the effects of a Coronal Mass Ejection be magnified in the South Atlantic Anomaly area?
During strong storms (which are generally caused by very strong/fast CMEs), yes the particle fluxes in the south Atlantic anomaly could be enhanced. Interestingly, the fluxes would be enhanced even if the anomaly were on the night-side during the storm.