I can't speak for the the PCE converter, but in my experience some converters do not work with digital only arcade sticks, they were designed for use with the dual shock and expects an analog input. This also means if using the Dual Shock 2 it might not work until you hit the analog button.
That being said, I have a Neo Geo Tototek converter and it didn't have any trouble with a PS1 joystick.