The triac in an SSR leaks very little, however there is additional circuitry for triggering the SSR that draws a small amount of current, and this is why you can detect some voltage when you connect a high resistance load such as a meter. You can find the leakage current on the datasheet for the SSR, but 2 or 3mA is typical. For typical high-current loads, the leakage is not a problem. If you use the SSR to drive a high power 240V 3-phase relay, then I suspect that this small amount of current will not be a problem. You can confirm that by checking how much current the big relay coil takes, which will be specified on its datasheet.
btw you almost certainly don't need the 200 ohm series resistor, because this is normally built into the SSR. If the input is specified as e.g. "3 to 20V" then it has a series resistor. If it didn't include a series resistor, then it would specify an operating current range, along with a typical forward voltage, just like a LED.
EDIT: That Sharp SSR has an unusually low off-state current, the datasheet says 100uA maximum. You can safely ignore it for the purposes of driving another relay. And (unusually for an SSR) it DOES need the series resistor.