Adding the extra wood will give you a complete surface for the tape to sit against, ensuring a good seal. The corners can be sealed with silicone (from the inside of the box) before the door goes on.Do I need the extra wood? Things might get tight in mounting the drivers.Paul
It looks tight, but you can tilt the drivers as they go in. With careful checking before you glue, you should be able to get a wide enough piece top and bottom to get your screws into.