If you manage to connect ps 2 controller to pc, as ps controller it should work. I am playing on ps4 controller with ds4 Windows and have no problems. Kind of wooden controlls but this is demake and it's damn great.

It works, try playing BBPSX as Non-steam game on steam, set up your controller configs, enter Big Screen Mode (I'm not sure, i think it just work without Big Screen Mode). Voila, you can use native PS1 buttons. 

My PS4 Controller wouldn't work, but I used Joy2Key to have it interpreted as a keyboard and it works great. You can probably do the same thing with your PS2 controller and Joy2Key is free.