Yes, you can just cut the 4 hole plug off and wire nut the wires to the cord wires. By folding the cord I just bend it tight to itself in a U shape. It'll do the same as twisting the sleeve. It just depends on the type of extension cord you get. Basically just try not to cut the wires.
Upon closer inspect that 4 hole plug is actually just snapped on to a 2 hole plug. It says Luminaire Disconnect. Should I still chop off the 2 hole plug?