A Q-logic bandpass is an off-the-shelf product that is not tuned for any particular woofer. The port or ports are press fit so the box is not airtight. Its untreated MDF and it probably won't hold up for a single season. Its not surprising that you're disappointed.
Your Sony amplifier will work fine. I would stay with woofers that are on the efficient side. A 10-inch in a ported enclosure will be the most bang for the buck and you can keep the box under 1.25 cu. ft.
A quality, custom-built, correctly tuned, waterproof box with terminal and poly mounting pads is around $200. A number of woofers from Kicker, Polk, JL, etc. will get the job done in the $100 to $200 range. So, $300 to $400 total.
Also, invest in a $20.00 set of subsonic filters if your amplifier doesn't have it built in.
The correct system tuning is as impactful as anything else.