Your inverter doesn't have a "low voltage disconnect" or perhaps one that runs them down too far before the cut-out. 20V is far too low if drawing only stand-by current from the batteries. Unfortunately, when using the inverter at high load, the voltage can drop significantly. In that case, a low-voltage disconnect shouldn't prevent the inverter from running at full power. This "either-or" condition makes it difficult to include intelligent low-voltage disconnect in a cheap inverter.
The gadget you posted - good grief it's just a voltage-controlled switch isn't it? From all the BS posted below the e-bay ad, it looks like they've cooked a few and posted the short-circuit inrush current as if that's a power rating. It might have a decent relay that can handle 100 amps or so. If you can get an independent source to confirm that the relay can handle switching the load your inverter can pull, then sure you can give it a try.
It's a fairly brutal way to control an inverter though. As mentioned above, using this as a LVD can cause the relay to trip because you're pulling a high current, not just because your wind source has stopped or the sun has gone down. Depends on the V setting you choose, and where it measures the system circuit voltage.