I have a generic PS2 controller (Shawain I think it comes up as). When I first connect it via USB it works as expected and when disconnected and connected via Bluetooth all good. However after pairing this way and the controller powers off or I turn off my Pi, when I press the PS2 button to get it to reconnect it is not recognized as a PlayStation controller and the button mapping is all wrong. When I connect it back with the USB it works as expected with Bluetooth but this is a little annoying to have to do each time
I have had nothing but problems using knock-off playstation controllers, both ps2 and ps3. They are really just garbage.
You are much better off buying an inexpensive XInput controller.
For versatility, Logitech controllers can switch from Xinput to Dinput.