Most LED's operate best at 15-24". So that leaves you 1' of space.
Maybe a 90w UFO with 120-150˚ lensing ? Autoflower ?
I would consider a larger cab but give it a go. Use the fixture as a ventilation fan. Mount the side holes inside the cab, top holes out the top of the fridge. Simply move the Plants up and down as needed.
JH
A $200 budget for a led grow light really limits your options other than low quality direct from china lights.