At the time of writing, Raspian is running Linux kernel 4.1.19 that has a bug in the bluetooth driver for Sony DualShock 4 controllers with newer firmware. Kernels as of 4.5rc1 have the fix. Until Raspian kernel packages catch up the solutions I can see are to backport this fix by patching and compiling a custom 4.1.x kernel or just running a DS4 of an older firmware.

Info on the driver fix is here:

https://git.kernel.org/cgit/linux/kernel/git/jikos/hid.git/commit/?h=for-4.5/sony&id=b71b5578a84d297954e4812ba0ca2d466e61cf42

Kernel Building on Raspberry Pi:

https://www.raspberrypi.org/documentation/linux/kernel/building.md