400 bucks for a shallow mount 12w3? No thanks.
I'm not familiar with 'Stereo Integrity,' but I'm guessing it's another upstart company selling their own designs direct to customers. Like RE, AA, and others...before they switched to selling to dealers only and the prices of their products doubled (or more).
Such companies usually offer some smokin' deals on very competently-designed drivers. Better built than JL's (less penny-pinching on simple stuff like cone and surround materials, for one) and you get more for the money (or 'same for less'). The linked SI sub is about the same as that JL in both displacement and power-handling.
I would only go to shallow subs as a last resort, though. If OP's max dimensions are right...assuming he's going off the outside and using 3/4" MDF...he should have 5-6" of mounting depth to play with.
Front face of box counts towards depth of sub, so that's 4.25" at the top of the box and 7.25" at the bottom. It 20" high, that means he can position his sub low, with the uppermost edge of the magnet coming in somewhere around or below the middle point between his 5" and 8" measurements- i.e. ~5.75" of depth, maybe more depending on the width of the motor and how much the angle of the box's face affects it.
I would pick this $79 sub over a JL w3.
http://www.woofersetc.com/p-8143-re...bwoofer-dual-4-ohm-oem-by-image-dynamics.aspx
It is the same thing as this, which is still a better deal than JL at $159.
http://www.woofersetc.com/p-4995-id12-v3-d4-image-dynamics-12-dual-4-ohm-v3-subwoofer.aspx
Mounting depth on an ID12 is 6". The other one is 5.75" because they omit a silly little plastic cover over the magnet. Cut the hole in your box at a point where the basket is pretty much butting up against the bottom of the box, and it should fit. Remember, you'd only need an actual 5" between the inner walls of your box, which is exactly what you would have at the halfway point between your 5" (3.5" inside) top and 8" (6.5") bottom.