1/4" hardi-backer board would be sufficient. I personally would not go without the cement board. The lids on these things get pretty damn heavy and lifting the lid from the front could cause a little flex in the plywood and crack out the grout lines over time. Just use 1/4" notched trowel to put down a base of mortar, cut the cement board to size and screw it down every 6 to 8 inches in the field of the board. This will provide the rigidity needed to prevent tile/grout damage and will help to provide another barrier of liquid protection in case of spillage. My 2 cents. Good luck!
I probably would have gone without the lid and just built the box with a plywood top and hardy board backing. What you have will work but it will be a bit heavier.