I happen to also have a sixaxis controller, and originally had your same problem.
When the device is plugged in USB mode, it behaves as expected, and most games will recognize it as the correct device. On Bluetooth, it works under a different spec, which most games don't support very well on Linux.
The basic problem here is that when the controller is on USB, it is registered to the Linux kernel as a HID device with all it's codes such as product and manufacturer IDs. This allows games to quickly and accurately identify the controller and load it's configs.
When you connect it via bluetooth, whichever bluetooth driver you are using attempts to connect the device as a HID via the Xserver or userspace desktop manager. This obscures much of the information about the device from the game or application, and since Linux support is commonly left in the dust compared to Windows, the game devs or engine don't spend the time to add this level of compatibility since it's not very well standardized.
Since the "joystick" to the game is now a virtual device instead of a USB one, they will be recognized as completely different devices. If the game supports it, you could add the custom config with the remapped controls to the game.
You can remap the joystick controls via a handy gui, perhaps
jscal may work. You could also try other non-official tools to remap the joystick, although I have yet to find any which work perfectly over bluetooth.
xboxdrv could be used to remap the controller to Xbox, though this solution probably isn't the easiest.
Another solution would be to look into other Bluetooth drivers (though I don't recommend this) that have better support. Personally, I've found the program
blueman to be extremely useful without having to reinstall bluetooth drivers.
Hope my experience helps.