My subwoofer suggestion for a budget system is: none. Don’t do it. Especially not for $2-300. In that budget range you can buy Townshend Pods, Cable Cradles, Purple Fuse, HFT, or any of a number of interconnects, power cords or even speaker cables that will have a bigger and entirely much more positive impact. Or you can get one seriously compromised and flawed sub that will actually make your system worse overall, owing to adding a cheap electronic circuit into the signal path, and bass that draws attention to itself.
The art of system building is compromise. This is even more important at the budget level. The two things that cost the most to do well are high volume and low bass. A budget system willing to compromise on high volume and omit the bottom octave can be incredibly captivating. Or you can have one that plays loud and goes deep but is utterly ordinary in every other way.
That’s the reality of the choices when you are on a budget. Which to give you some idea, I didn’t have a DBA until after Moab. So I would be in no hurry. My two bits.