Wine mini fridges are fine, I've been using one for years. It'll cold crash into the high 30's no problem. Just make sure it has a compressor and not a peltier cooling device. If it has a big "hump" or shelf inside the fridge, that's the compressor compartment.
If it has electronic controls, all you have to do is disconnect the compressor wires going to the control board, and wire them to your own controller (which may be as simple as splicing in an appropriate 120V 3 prong extension cord with the female end cut off). You can leave the old control board in place, which may have switches for lights, a temp readout, etc. Your fridge will have now have two power cords, the compressor one goes to your controller, the other one can be plugged into the wall.