I finally got bluetooth working on my GPD Pocket.
This is with Fedora 26 and the rawhide kernel, but it probably works with many other kernels.
/sys/kernel/debug/usb/devices tells me the bt chip is a Broadcom Corp BCM2045A0
I thought this would need special firmware etc, but that's not the case or the firmware is already available on Fedora.
The issue here is the weird USB id: 0000:0000
To get it working:
# modprobe btusb
# echo "0000 0000" > /sys/bus/usb/drivers/btusb/new_id
To make this permanent (On Fedora 26):
Save this to /etc/rc.d/rc.local:
echo "0000 0000" > /sys/bus/usb/drivers/btusb/new_id
# chmod 755 /etc/rc.d/rc.local
# systemctl enable rc-local