As per the Custom Button Mapping guide, once mappings are saved they should be switchable via "clicking in on the left thumbstick" or by holding down "L and R then cycle through using the left and right on the D pad."
Switching Mapping layers does not seem to work on my DS4 controller.
Can confirm button binding changes made to mapping layer 1 are saved correctly using PSP Bluetooth Version release-2.3.2. Tested in XMB and game. Changes to DS4 light bar also work.