How do you attach this fence panel to a 4x4 post?
The hanger you have pictured is not correct for the panel you have pictured as the 2x4 is not oriented correctly for them.
 If you are not comfortable using just deck screws you could put an L bracket on the bottom of the panel to carry the weight of it and use a 1/4" lag bolt or carriage bolt at the top for lateral or shear strength