Haven't experimented much, but so far I am feeling the 8bitdo is a flaming piece of garbage on the psx mini. It puts a mouse cursor on top of the screen. Only way to remove I find is unplugging and plugging the unit back in and resyncing the controller (DS4). Imagine having to do that every time you leave or enter a game. There might be a proper way to configure the controller, but I haven't figured it out yet. Haven't gotten analog or rumble on bleemsync or retroarch yet. Just a faux analog where it reads the thumbstick as the d-pad. It also leaves power on even when the unit is off. When you shut down the console, it does send a power off command to the dualshock, but if you accidentally press any button on the dualshock, it will connect back to the 8bitdo and stay on, even with the unit off.
Still need to experiment more, but my initial thoughts are bad. Might as well just plug in the controller directly with the USB port.
Thing is, for all that I see, at least retroarch should support analog and rumble. The issues are likely in the 8bitdo.