It should work with any powered sub whose crossover control acts as a low-pass filter on the speaker-level inputs - that is to say, it will allow the sub to play everything up to the "crossover" point (which is actually a narrow range, rather than a fixed point), and nothing above it.
In your scenario:
- You sent the full range of signal to the speakers and the full range of signal to the sub.
- You use the sub's crossover control to blend the sub with the speakers by raising it until it overlaps smoothly with the point where your speakers low-frequency extension tapers off.
between those 2? I'd get the second one, the 10 incher... but only if those were my only 2 options. There are certainly much better subs out there, but staying close to the $100 mark, and being small.... I don't have any real experience with any better within those criteria.