I think you can use a Chronus Max adapter to use 360 controllers on PS4, but I haven't heard that many positive things about it from the competitive scene.
Other option would be to gut a Hori Fighting Commander 4 and wire the PCB into your stick.
Yeah you can, but if you go that route get a TitanOne as it's the official/newer model. The problem with it is that the PS4 checks controllers for authentication every 10 minutes, the TitanOne has an option to automatically reconnect the controller during those periods. I wouldn't recommend it for fighting games because the moment it reconnects, it makes you lose control of gamepaly for a second. It's fine for most other games if you're just playing casually because it's so quick but fighting games IMO is a different ballgame. Hopefully they can fix that completely. Also if you don't have a PC to program the device you're out of luck as well.