The report descriptor actually describes a lot more report-IDs than
just "0x01" which you describe in hidraw.txt. Are you sure the
report-descriptors are the same for USB and BT?

The report descriptor when connected via BT is different, or at least what
I found at '/sys/bus/hid/devices/0005\:054C\:05C4.0006/report_descriptor'.

Unfortunately I wasn't able to decode it.
--
simon@womble:~/dualshock4$ ~/hidrd-0.2.0/src/hidrd-convert des_bt.bin -o code
Failed to read input item at offset 356:
invalid item encountered
--

Thanks for the other suggestions.
Simon