Putting one in the cabinet will cause the glass to vibrate and make horrible noises.

The thing I dont like about the Flipper Fidelity kit is that it really taxes the games amp that was not designed to push that much power. I heard a fun house that had the speakers installed and the amp was getting so hot it was shutting down. The reccomend you use their speakers with a Pinsound board. Well all said and done your going to have +$400 into it, or you could just spend the 100 on the external sub and throw it under the caibnet.

Or you can take that sub apart and use its amp and speaker in your cabinet. No guarantee it will sound any good but you could do it.

You could box in the sub in the cabinet to get it tuned if there is room. or you could attach the complete sub to the bottom of the cabinet so its off the floor.