How "hard" is your 12" limit, free diver? An SVS' SB1000 - a sealed sub rated into mid 20 Hz - is right around 13" cubed. It has speaker in and out as well as line level connections. Price is also in line with your budget. Worth noting, the next SVS model up has line level connections only.
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Hsu 12", $512
Having said this, without EQ, smaller subs are better. Maybe even a 10" such as this one.
Subwoofer's are not plug and play. The deeper they go the more likely to run into room interaction issues. Placement, EQ and bass traps are often needed. If none of those are items you can spend time and money on, smaller is safer.