As the PlayStation Classic's Controller could be used on PC as a Generic Controller. I am thinking may be the reverse is also true, i.e. a Generic PC Joystick Controller like Saga Virtual Stick (CS2-0007) could be used on PlayStation Classic without modding. (Together with the autofire feature)
I does not have my PlayStation Classic console with me, so couldn't try.
*Meow* I am as serious as a cat could possible be. *Meow*
Without soft-modding (which you should definitely do if only for better PS1 emulation), the Brook PS3/4 to PS1/2 Super Converter will allow you to use PS3/PS4 controllers on the PS Classic, with it you can use a supported Arcade Stick with a stock PS Classic. There is a micro to UBS-A cable you use to connect the converter to the PS Classic and then you plug your Arcade Stick into the Brook Converters female USB-A Port.
I have Autobleem on mine and can plug in my HORI RAP 4 Kai directly and map buttons without the need of the Brook Adapter.