I call BS on the cheapo grid-tie inverter. About the only thing it can possibly do is pass AC through the capacitor to the battery. This will just alternately charge and discharge the battery with AC current. Since there is no current limiting other than the impedance of the capacitor, it could (and probably will) damage it.
If you have built one of these contraptions, measure the actual power drawn from the AC line. Then measure the true RMS current into and out of the battery. To see the actual power to and from the battery, you could use a wattmeter, or you could measure the DC current to or from the battery and multiply by the battery voltage. But unless the capacitor is shorted, or still charging, the DC current will be zero.
seems to show real grid tie inverters, at a more realistic price of about $1/watt. When I first saw the links, I thought it was for "m
", which you would need to get the cheapo inverter working.